Method and System of Computing and Rendering the Nature of the Excited Electronic States of Atoms and Atomic Ions

ABSTRACT

A method and system of physically solving the charge, mass, and current density functions of excited-state atoms and atomic ions using Maxwell&#39;s equations and computing and rendering the nature of excited-state electrons using the solutions. The results can be displayed on visual or graphical media. The display can be static or dynamic such that electron spin and rotation motion can be displayed in an embodiment. The displayed information is useful to anticipate reactivity and physical properties. The insight into the nature of excited-state electrons can permit the solution and display of those of other atoms and atomic ions and provide utility to anticipate their reactivity and physical properties as well as spectral absorption and emission to lead to new optical materials and light sources.

I. INTRODUCTION

1. Field of the Invention

This invention relates to a method and system of physically solving thecharge, mass, and current density functions of excited states of atomsand atomic ions and computing and rendering the nature of these speciesusing the solutions. The results can be displayed on visual or graphicalmedia. The displayed information is useful to anticipate reactivity andphysical properties. The insight into the nature of excited-stateelectrons can permit the solution and display of other excited-stateatoms and ions and provide utility to anticipate their reactivity,physical properties, and spectral absorption and emission.

Rather than using postulated unverifiable theories that treat atomicparticles as if they were not real, physical laws are now applied toatoms and ions. In an attempt to provide some physical insight intoatomic problems and starting with the same essential physics as Bohr ofthe e⁻ moving in the Coulombic field of the proton with a true waveequation as opposed to the diffusion equation of Schrödinger, aclassical approach is explored which yields a model which is remarkablyaccurate and provides insight into physics on the atomic level. Theproverbial view deeply seated in the wave-particle duality notion thatthere is no large-scale physical counterpart to the nature of theelectron is shown not to be correct. Physical laws and intuition may berestored when dealing with the wave equation and quantum atomicproblems.

Specifically, a theory of classical quantum mechanics (CQM) was derivedfrom first principles as reported previously [reference Nos. 1-7] thatsuccessfully applies physical laws to the solution of atomic problemsthat has its basis in a breakthrough in the understanding of thestability of the bound electron to radiation. Rather than using thepostulated Schrödinger boundary condition: “Ψ→0 as r→∞”, which leads toa purely mathematical model of the electron, the constraint is based onexperimental observation. Using Maxwell's equations, the classical waveequation is solved with the constraint that the bound n=1-state electroncannot radiate energy. Although it is well known that an acceleratedpoint particle radiates, an extended distribution modeled as asuperposition of accelerating charges does not have to radiate. A simpleinvariant physical model arises naturally wherein the predicted resultsare extremely straightforward and internally consistent requiringminimal math as in the case of the most famous equations of Newton,Maxwell, Einstein, de Broglie, and Planck on which the model is based.No new physics is needed; only the known physical laws based on directobservation are used. The solution of the excited states of one-electronatoms is given in R. Mills, The Grand Unified Theory of ClassicalQuantum Mechanics, January 2005 Edition, BlackLight Power, Inc.,Cranbury, N.J., (“'05 Mills GUT”) which is herein incorporated byreference. This Invention further comprises the accurate solution of thehelium-atom excited states which provides a physical algorithm to solvethe excited states of other multi-electron atoms.

2. Background of the Invention

2A. Classical Quantum Theory of the Atom Based on Maxwell's Equations

The old view that the electron is a zero or one-dimensional point in anall-space probability wave function Ψ(x) is not taken for granted. Thetheory of classical quantum mechanics (CQM), derived from firstprinciples, must successfully and consistently apply physical laws onall scales [1-7]. Stability to radiation was ignored by all past atomicmodels. Historically, the point at which QM broke with classical lawscan be traced to the issue of nonradiation of the one electron atom.Bohr just postulated orbits stable to radiation with the furtherpostulate that the bound electron of the hydrogen atom does not obeyMaxwell's equations—rather it obeys different physics [1-10]. Laterphysics was replaced by “pure mathematics” based on the notion of theinexplicable wave-particle duality nature of electrons which lead to theSchrödinger equation wherein the consequences of radiation predicted byMaxwell's equations were ignored. Ironically, Bohr, Schrödinger, andDirac used the Coulomb potential, and Dirac used the vector potential ofMaxwell's equations. But, all ignored electrodynamics and thecorresponding radiative consequences. Dirac originally attempted tosolve the bound electron physically with stability with respect toradiation according to Maxwell's equations with the further constraintsthat it was relativistically invariant and gave rise to electron spin[11]. He and many founders of QM such as Sommerfeld, Bohm, and Weinsteinwrongly pursued a planetary model, were unsuccessful, and resorted tothe current mathematical-probability-wave model that has many problems[10, 11-14]. Consequently, Feynman for example, attempted to use firstprinciples including Maxwell's equations to discover new physics toreplace quantum mechanics [15].

Physical laws may indeed be the root of the observations thought to be“purely quantum mechanical”, and it was a mistake to make the assumptionthat Maxwell's electrodynamic equations must be rejected at the atomiclevel. Thus, in the present approach, the classical wave equation issolved with the constraint that a bound n=1−state electron cannotradiate energy.

Herein, derivations consider the electrodynamic effects of movingcharges as well as the Coulomb potential, and the search is for asolution representative of the electron wherein there is acceleration ofcharge motion without radiation. The mathematical formulation for zeroradiation based on Maxwell's equations follows from a derivation by Haus[16]. The function that describes the motion of the electron must notpossess spacetime Fourier components that are synchronous with wavestraveling at the speed of light. Similarly, nonradiation is demonstratedbased on the electron's electromagnetic fields and the Poynting powervector.

It was shown previously [1-7] that CQM gives closed form solutions forthe atom including the stability of the n=1 state and the instability ofthe excited states, the equation of the photon and electron in excitedstates, the equation of the free electron, and photon which predict thewave particle duality behavior of particles and light. The current andcharge density functions of the electron may be directly physicallyinterpreted. For example, spin angular momentum results from the motionof negatively charged mass moving systematically, and the equation forangular momentum, r×p, can be applied directly to the wave function (acurrent density function) that describes the electron. The magneticmoment of a Bohr magneton, Stern Gerlach experiment, g factor, Lambshift, resonant line width and shape, selection rules, correspondenceprinciple, wave particle duality, excited states, reduced mass,rotational energies, and momenta, orbital and spin splitting,spin-orbital coupling, Knight shift, and spin-nuclear coupling, andelastic electron scattering from helium atoms, are derived inclosed-form equations based on Maxwell's equations. The calculationsagree with experimental observations.

The Schrödinger equation gives a vague and fluid model of the electron.Schrödinger interpreted eΨ*(x)Ψ(x) as the charge-density or the amountof charge between x and x+dx (Ψ* is the complex conjugate of Ψ).Presumably, then, he pictured the electron to be spread over largeregions of space. After Schrödinger's interpretation, Max Born, who wasworking with scattering theory, found that this interpretation led toinconsistencies, and he replaced the Schrödinger interpretation with theprobability of finding the electron between x and x+dx as∫Ψ(x)Ψ*(x)dx  (1)Born's interpretation is generally accepted. Nonetheless, interpretationof the wave function is a never-ending source of confusion and conflict.Many scientists have solved this problem by conveniently adopting theSchrödinger interpretation for some problems and the Born interpretationfor others. This duality allows the electron to be everywhere at onetime-yet have no volume. Alternatively, the electron can be viewed as adiscrete particle that moves here and there (from r=0 to r=∞), and ΨΨ*gives the time average of this motion.

In contrast to the failure of the Bohr theory and the nonphysical,adjustable-parameter approach of quantum mechanics, multielectron atoms[1, 5] and the nature of the chemical bond [1, 4] are given by exactclosed-form solutions containing fundamental constants only. Using thenonradiative wave equation solutions that describe the bound electronhaving conserved momentum and energy, the radii are determined from theforce balance of the electric, magnetic, and centrifugal forces thatcorresponds to the minimum of energy of the system. The ionizationenergies are then given by the electric and magnetic energies at theseradii. The spreadsheets to calculate the energies from exact solutionsof one through twenty-electron atoms are given in '05 Mills GUT [1] andare available from the internet [17]. For 400 atoms and ions theagreement between the predicted and experimental results is remarkable.

The background theory of classical quantum mechanics (CQM) for thephysical solutions of atoms and atomic ions is disclosed in R. Mills,The Grand Unified Theory of Classical Quantum Mechanics, January 2000Edition, BlackLight Power, Inc., Cranbury, N.J., (“'00 Mills GUT”),provided by BlackLight Power, Inc., 493 Old Trenton Road, Cranbury,N.J., 08512; R. Mills, The Grand Unified Theory of Classical QuantumMechanics, September 2001 Edition, BlackLight Power, Inc., Cranbury,N.J., Distributed by Amazon.com (“'01 Mills GUT”), provided byBlackLight Power, Inc., 493 Old Trenton Road, Cranbury, N.J., 08512; R.Mills, The Grand Unified Theory of Classical Quantum Mechanics, July2004 Edition, BlackLight Power, Inc., Cranbury, N.J., (“'04 Mills GUT”),provided by BlackLight Power, Inc., 493 Old Trenton Road, Cranbury,N.J., 08512; R. Mills, The Grand Unified Theory of Classical QuantumMechanics, January 2005 Edition, BlackLight Power, Inc., Cranbury, N.J.,(“'05 Mills GUT”), provided by BlackLight Power, Inc., 493 Old TrentonRoad, Cranbury, N.J., 08512 (posted at www.blacklightpower.com); inprior PCT applications PCT/US02/35872; PCT/US02/06945; PCT/US02/06955;PCT/US01/09055; PCT/US01/25954; PCT/US00/20820; PCT/US00/20819;PCT/US00/09055; PCT/US99/17171; PCT/US99/17129; PCT/US 98/22822;PCT/US98/14029; PCT/US96/07949; PCT/US94/02219; PCT/US91/08496;PCT/US90/01998; and PCT/JS89/05037 and U.S. Pat. No. 6,024,935; theentire disclosures of which are all incorporated herein by reference;(hereinafter “Mills Prior Publications”).

II. SUMMARY OF THE INVENTION

An object of the present invention is to solve the charge (mass) andcurrent-density functions of excited-state atoms and atomic ions fromfirst principles. In an embodiment, the solution for the excited andnon-excited state is derived from Maxwell's equations invoking theconstraint that the bound electron before excitation does not radiateeven though it undergoes acceleration.

Another objective of the present invention is to generate a readout,display, or image of the solutions so that the nature of excited-stateatoms and atomic ions can be better understood and potentially appliedto predict reactivity and physical and optical properties.

Another objective of the present invention is to apply the methods andsystems of solving the nature of excited-state electrons and theirrendering to numerical or graphical form to all atoms and atomic ions.

Bound electrons are described by a charge-density (mass-density)function which is the product of a radial delta function(ƒ(r)=δ(r−r_(n))), two angular functions (spherical harmonic functions),and a time harmonic function. Thus, a bound electron is a dynamic“bubble-like” charge-density function. The two-dimensional sphericalsurface called an electron orbitsphere shown in FIG. 1 can exist in abound state at only specified distances from the nucleus. Moreexplicitly, the orbitsphere comprises a two-dimensional spherical shellof moving charge. The current pattern of the orbitsphere that gives riseto the phenomenon corresponding to the spin quantum number comprises aninfinite series of correlated orthogonal great circle current loops. Asgiven in the Orbitsphere Equation of Motion for l=0 section of '05 MillsGUT [1], the current pattern (shown in FIG. 2) is generated over thesurface by two orthogonal sets of an infinite series of nested rotationsof two orthogonal great circle current loops where the coordinate axesrotate with the two orthogonal great circles. Each infinitesimalrotation of the infinite series is about the new x-axis and new y-axiswhich results from the preceding such rotation. For each of the two setsof nested rotations, the angular sum of the rotations about eachrotating x-axis and y-axis totals √{square root over (2)}π radians. Thespin function of the electron corresponds to the nonradiative n=1, l=0state which is well known as an s state or orbital. (See FIG. 1 for thecharge function and FIG. 2 for the current function.) In cases oforbitals of excited states with the l quantum number not equal to zeroand which are not constant as given by Eq. (1.64) of Ref. [1], theconstant spin function is modulated by a time and spherical harmonicfunction as given by Eq. (1.65) of Ref. [1] and shown in FIG. 3. Themodulation or traveling charge-density wave corresponds to an orbitalangular momentum in addition to a spin angular momentum. These statesare typically referred to as p, d, f, etc. orbitals.

Each orbitsphere is a spherical shell of negative charge (totalcharge=−e) of zero thickness at a distance r_(n) from the nucleus(charge=+Ze). It is well known that the field of a spherical shell ofcharge is zero inside the shell and that of a point charge at the originoutside the shell [1] (See FIG. 1.12 of Ref. [1]). The field of eachelectron can be treated as that corresponding to a −e charge at theorigin with $E = \frac{- e}{4\pi\quad ɛ_{o}r^{2}}$for r>r_(n) and E=0 for r<r_(n) where r_(n) is the radius of theelectron orbitsphere. Thus, as shown in the Two-Electron Atom section of'05 Mills GUT [1], the central electric fields due to the helium nucleusare $E = \frac{2e}{4\pi\quad ɛ_{o}r^{2}}$and $E = \frac{e}{4\pi\quad ɛ_{o}r^{2}}$for r<r₁ and r₁<r<r₂, respectively. In the ground state of the heliumatom, both electrons are at r₁=r₂=0.567α_(o). When a photon is absorbed,one of the initially indistinguishable electrons called electron 1 movesto a smaller radius, and the other called electron 2 moves to a greaterradius. In the limiting case of the absorption of an ionizing photon,electron 1 moves to the radius of the helium ion, r₁=0.5α_(o), andelectron 2 moves to a continuum radius, r₂=∞. When a photon is absorbedby the ground state helium atom it generates an effective charge,Z_(P-eff), within the second orbitsphere such that the electrons move inopposite radial directions while conserving energy and angular momentum.We can determine Z_(P-eff) of the “trapped photon” electric field byrequiring that the resonance condition is met for photons of discreteenergy, frequency, and wavelength for electron excitation in anelectromagnetic potential energy well.

In contrast to the shortcomings of quantum mechanics, with classicalquantum mechanics (CQM), all excited states of the helium atom can beexactly solved in closed form. Photon absorption occurs by an excitationof a Maxwellian multipole cavity mode wherein the excitation isquantized according to the quantized energy and angular momentum of thephoton given by

w and

, respectively. The photon quantization causes the centralelectric-field corresponding the superimposed fields of the nucleus,electron 1, and the photon to be quantized and of magnitude of areciprocal integer times that of the proton. This field and thephase-matched angular dependence of the trapped photon and excited-stateelectron as well as the spin orientation of the excited-state electrondetermine the central forces. The radii of electron 2 are determinedfrom the force balance of the electric, magnetic, and centrifugal forcesthat corresponds to the minimum of energy of the system. Since themagnetic energies are relatively insignificant, in one embodiment, theexcited state energies are then given by one physical term in each case,the Coulombic energy at the calculated radius. In additionalembodiments, additional small terms may refine the solutions. Given thetypical average relative difference is about 5 significant figures whichis within the error of the experimental data, this result is remarkableand strongly confirms that the physical CQM solution of helium iscorrect.

The presented exact physical solutions for the excited states of thehelium atom can be applied to other atoms and ions to solve for theirexcited states. These solution can be used to predict the properties ofelements and ions and engineer compositions of matter in a manner whichis not possible using quantum mechanics. It also for the prediction ofthe spectral absorption and emission. This in term can be used todevelop new light filters or absorbers as well as new light sources suchas lasers, lamps, and spectral standards.

In an embodiment., the physical, Maxwellian solutions for the dimensionsand energies of excited-state atom and atomic ions are processed with aprocessing means to produce an output. Embodiments of the system forperforming computing and rendering of the nature of the excited-stateatomic and atomic-ionic electrons using the physical solutions maycomprise a general purpose computer. Such a general purpose computer mayhave any number of basic configurations. For example, such a generalpurpose computer may comprise a central processing unit (CPU), one ormore specialized processors, system memory, a mass storage device suchas a magnetic disk, an optical disk, or other storage device, an inputmeans such as a keyboard or mouse, a display device, and a printer orother output device. A system implementing the present invention canalso comprise a special purpose computer or other hardware system andall should be included within its scope.

III. BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows the orbitsphere in accordance with the present inventionthat is a two dimensional spherical shell of zero thickness with theBohr radius of the hydrogen atom, r=α_(H).

FIG. 2 shows the current pattern of the orbitsphere in accordance withthe present invention from the perspective of looking along the z-axis.The current and charge density are confined to two dimensions at r_(n)32 nr₁. The corresponding charge density function is uniform.

FIG. 3 shows that the orbital function modulates the constant (spin)function (shown for t=0; three-dimensional view).

FIG. 4 shows the normalized radius as a function of the velocity due torelativistic contraction.

FIG. 5 shows the magnetic field of an electron orbitsphere (z-axisdefined as the vertical axis).

FIG. 6 shows a plot of the predicted and experimental energies of levelsassigned by NIST, and

FIG. 7 shows a computer rendering of the helium atom in the n=2, l=1excited state according to the present Invention.

IV. DETAILED DESCRIPTION OF THE INVENTION

The following preferred embodiments of the invention disclose numerouscalculations which are merely intended as illustrative examples. Basedon the detailed written description, one skilled in the art would easilybe able to practice this Invention within other like calculations toproduce the desired result without undue effort.

1. One-Electron Atoms

1. One-Electron Atoms

One-electron atoms include the hydrogen atom, He⁺, Li²⁺, Be³⁺, and soon. The mass-energy and angular momentum of the electron are constant;this requires that the equation of motion of the electron be temporallyand spatially harmonic. Thus, the classical wave equation applies and$\begin{matrix}{{\left\lbrack {{\nabla^{2}{- \frac{1}{v^{2}}}}\frac{\partial^{2}}{\partial t^{2}}} \right\rbrack{\rho\left( {r,\theta,\phi,t} \right)}} = 0} & (2)\end{matrix}$where ρ(r,θ,φ,t) is the time dependent charge density function of theelectron in time and space. In general, the wave equation has aninfinite number of solutions. To arrive at the solution which representsthe electron, a suitable boundary condition must be imposed. It is wellknown from experiments that each single atomic electron of a givenisotope radiates to the same stable state. Thus, the physical boundarycondition of nonradiation of the bound electron was imposed on thesolution of the wave equation for the time dependent charge densityfunction of the electron [1-3, 5]. The condition for radiation by amoving point charge given by Haus [16] is that its spacetime Fouriertransform does possess components that are synchronous with wavestraveling at the speed of light. Conversely, it is proposed that thecondition for nonradiation by an ensemble of moving point charges thatcomprises a current density function is

-   -   For non-radiative states, the current-density function must NOT        possess spacetime    -   Fourier components that are synchronous with waves traveling at        the speed of light.        The time, radial, and angular solutions of the wave equation are        separable. The motion is time harmonic with frequency ω_(n). A        constant angular function is a solution to the wave equation.        Solutions of the Schrödinger wave equation comprising a radial        function radiate according to Maxwell's equation as shown        previously by application of Haus' condition [1]. In fact, it        was found that any function which permitted radial motion gave        rise to radiation. A radial function which does satisfy the        boundary condition is a radial delta function $\begin{matrix}        {{f(r)} = {\frac{1}{r^{2}}{\delta\left( {r - r_{n}} \right)}}} & (3)        \end{matrix}$        This function defines a constant charge density on a spherical        shell where r_(n)=nr₁ wherein n is an integer in an excited        state, and Eq. (2) becomes the two-dimensional wave equation        plus time with separable time and angular functions. Given time        harmonic motion and a radial delta function, the relationship        between an allowed radius and the electron wavelength is given        by        2πr_(n)=λ_(n)  (4)        where the integer subscript n here and in Eq. (3) is determined        during photon absorption as given in the Excited States of the        One-Electron Atom (Quantization) section of Ref. [1]. Using the        observed de Broglie relationship for the electron mass where the        coordinates are spherical, $\begin{matrix}        {\lambda_{n} = {\frac{h}{p_{n}} = \frac{h}{m_{e}v_{n}}}} & (5)        \end{matrix}$        and the magnitude of the velocity for every point on the        orbitsphere is $\begin{matrix}        {v_{n} = \frac{\hslash}{m_{e}r_{n}}} & (6)        \end{matrix}$        The sum of the |L_(i)|, the magnitude of the angular momentum of        each infinitesimal point of the orbitsphere of mass m_(i), must        be constant. The constant is        . $\begin{matrix}        {{\sum{L_{i}}} = {{\sum{{{r \times m_{i}}v}}} = {{m_{e}r_{n}\frac{\hslash}{m_{e}r_{n}}} = \hslash}}} & (7)        \end{matrix}$        Thus, an electron is a spinning, two-dimensional spherical        surface (zero thickness), called an electron orbitsphere shown        in FIG. 1, that can exist in a bound state at only specified        distances from the nucleus determined by an energy minimum. The        corresponding current function shown in FIG. 2 which gives rise        to the phenomenon of spin is derived in the Spin Function        section. (See the Orbitsphere Equation of Motion for l=0 of Ref.        [1] at Chp. 1.)

Nonconstant functions are also solutions for the angular functions. Tobe a harmonic solution of the wave equation in spherical coordinates,these angular functions must be spherical harmonic functions [18]. Azero of the spacetime Fourier transform of the product function of twospherical harmonic angular functions, a time harmonic function, and anunknown radial function is sought. The solution for the radial functionwhich satisfies the boundary condition is also a delta function given byEq. (3). Thus, bound electrons are described by a charge-density(mass-density) function which is the product of a radial delta function,two angular functions (spherical harmonic functions), and a timeharmonic function. $\begin{matrix}{{{\rho\left( {r,\theta,\phi,t} \right)} = {{{f(r)}{A\left( {\theta,\phi,t} \right)}} = {\frac{1}{r^{2}}{\delta\left( {r - r_{n}} \right)}{A\left( {\theta,\phi,t} \right)}}}};{{A\left( {\theta,\phi,t} \right)} = {{Y\left( {\theta,\phi} \right)}{k(t)}}}} & (8)\end{matrix}$In these cases, the spherical harmonic functions correspond to atraveling charge density wave confined to the spherical shell whichgives rise to the phenomenon of orbital angular momentum. The orbitalfunctions which modulate the constant “spin” function shown graphicallyin FIG. 3 are given in the Sec. 1.B.1.A. Spin Function

The orbitsphere spin function comprises a constant charge (current)density function with moving charge confined to a two-dimensionalspherical shell. The magnetostatic current pattern of the orbitspherespin function comprises an infinite series of correlated orthogonalgreat circle current loops wherein each point charge (current) densityelement moves time harmonically with constant angular velocity$\begin{matrix}{\omega_{n} = \frac{\hslash}{m_{e}r_{n}^{2}}} & (9)\end{matrix}$

The uniform current density function Y₀ ⁰(φ,θ), the orbitsphere equationof motion of the electron (Eqs. (14-15)), corresponding to the constantcharge function of the orbitsphere that gives rise to the spin of theelectron is generated from a basis set current-vector field defined asthe orbitsphere current-vector field (“orbitsphere-cvf”). This in turnis generated over the surface by two complementary steps of an infiniteseries of nested rotations of two orthogonal great circle current loopswhere the coordinate axes rotate with the two orthogonal great circlesthat serve as a basis set. The algorithm to generate the current densityfunction rotates the great circles and the corresponding x′y′z′coordinates relative to the xyz frame. Each infinitesimal rotation ofthe infinite series is about the new i′-axis and new j′-axis whichresults from the preceding such rotation. Each element of the currentdensity function is obtained with each conjugate set of rotations. InAppendix III of Ref. [1], the continuous uniform electron currentdensity function Y₀ ⁰(φ,θ) having the same angular momentum componentsas that of the orbitsphere-cvf is then exactly generated from thisorbitsphere-cvf as a basis element by a convolution operator comprisingan autocorrelation-type function.

For Step One, the current density elements move counter clockwise on thegreat circle in the y′z′-plane and move clockwise on the great circle inthe x′z′-plane. The great circles are rotated by an infinitesimal angle±Δα_(i) (a positive rotation around the x′-axis or a negative rotationabout the z′-axis for Steps One and Two, respectively) and then by±Δα_(j) (a positive rotation around the new y′-axis or a positiverotation about the new x′-axis for Steps One and Two, respectively). Thecoordinates of each point on each rotated great circle (x′,y′,z′) isexpressed in terms of the first (x,y,z) coordinates by the followingtransforms where clockwise rotations and motions are defined as positivelooking along the corresponding axis: $\begin{matrix}{{Step}\quad{One}} & \quad \\{\begin{bmatrix}x \\y \\z\end{bmatrix} = {{{{\begin{bmatrix}{\cos\left( {\Delta\quad\alpha_{y}} \right)} & 0 & {- {\sin\left( {\Delta\quad\alpha_{y}} \right)}} \\0 & 1 & 0 \\{\sin\left( {\Delta\quad\alpha_{y}} \right)} & 0 & {\cos\left( {\Delta\quad\alpha_{y}} \right)}\end{bmatrix}\begin{bmatrix}1 & 0 & 0 \\0 & {\cos\left( {\Delta\quad\alpha_{x}} \right)} & {\sin\left( {\Delta\quad\alpha_{x}} \right)} \\0 & {- {\sin\left( {\Delta\quad\alpha_{x}} \right)}} & {\cos\left( {\Delta\quad\alpha_{x}} \right)}\end{bmatrix}}\begin{bmatrix}x^{\prime} \\y^{\prime} \\z^{\prime}\end{bmatrix}}\begin{bmatrix}x \\y \\z\end{bmatrix}} = {\begin{bmatrix}{\cos\left( {\Delta\quad\alpha_{y}} \right)} & {{\sin\left( {\Delta\quad\alpha_{y}} \right)}{\sin\left( {\Delta\quad\alpha_{x}} \right)}} & {{- {\sin\left( {\Delta\quad\alpha_{y}} \right)}}{\cos\left( {\Delta\quad\alpha_{x}} \right)}} \\0 & {\cos\left( {\Delta\quad\alpha_{x}} \right)} & {\sin\left( {\Delta\quad\alpha_{x}} \right)} \\{\sin\left( {\Delta\quad\alpha_{y}} \right)} & {{- {\cos\left( {\Delta\quad\alpha_{y}} \right)}}{\sin\left( {\Delta\quad\alpha_{x}} \right)}} & {{\cos\left( {\Delta\quad\alpha_{y}} \right)}{\cos\left( {\Delta\quad\alpha_{x}} \right)}}\end{bmatrix}\begin{bmatrix}x^{\prime} \\y^{\prime} \\z^{\prime}\end{bmatrix}}}} & (10) \\{{Step}\quad{Two}} & \quad \\{{\begin{bmatrix}x \\y \\z\end{bmatrix} = {{{{\begin{bmatrix}1 & 0 & 0 \\0 & {\cos\left( {\Delta\quad\alpha_{x}} \right)} & {\sin\left( {\Delta\quad\alpha_{x}} \right)} \\0 & {- {\sin\left( {\Delta\quad\alpha_{x}} \right)}} & {\cos\left( {\Delta\quad\alpha_{x}} \right)}\end{bmatrix}\begin{bmatrix}{\cos\left( {\Delta\quad\alpha_{z}} \right)} & {\sin\left( {\Delta\quad\alpha_{z}} \right)} & 0 \\{- {\sin\left( {\Delta\quad\alpha_{z}} \right)}} & {\cos\left( {\Delta\quad\alpha_{z\quad}} \right)} & 0 \\0 & 0 & 1\end{bmatrix}}\begin{bmatrix}x^{\prime} \\y^{\prime} \\z^{\prime}\end{bmatrix}}\begin{bmatrix}x \\y \\z\end{bmatrix}} = {\begin{bmatrix}{\cos\left( {\Delta\quad\alpha_{z}} \right)} & {\sin\left( {\Delta\quad\alpha_{z}} \right)} & 0 \\{{- {\cos\left( {\Delta\quad\alpha_{x}} \right)}}{\sin\left( {\Delta\quad\alpha_{z}} \right)}} & {{\cos\left( {\Delta\quad\alpha_{x}} \right)}{\cos\left( {\Delta\quad\alpha_{z}} \right)}} & {\sin\left( {\Delta\quad\alpha_{x}} \right)} \\{{\sin\left( {\Delta\quad\alpha_{x}} \right)}{\sin\left( {\Delta\quad\alpha_{z}} \right)}} & {{- {\sin\left( {\Delta\quad\alpha_{x}} \right)}}{\cos\left( {\Delta\quad\alpha_{z}} \right)}} & {\cos\left( {\Delta\quad\alpha_{x}} \right)}\end{bmatrix}\begin{bmatrix}x^{\prime} \\y^{\prime} \\z^{\prime}\end{bmatrix}}}}{{{where}\quad{the}\quad{angular}\quad{sum}\quad{is}\quad{\lim\limits_{{\Delta\quad\alpha}\rightarrow 0}{\sum\limits_{n = 1}^{\Delta\quad\overset{\frac{\sqrt{2}}{2}\pi}{\alpha_{i,j}}}\quad{{\Delta\quad\alpha_{i,j}}}}}} = {\frac{\sqrt{2}}{2}{\pi.}}}} & (11)\end{matrix}$

The orbitsphere-cvf is given by n reiterations of Eqs. (10) and (11) foreach point on each of the two orthogonal great circles during each ofSteps One and Two. The output given by the non-primed coordinates is theinput of the next iteration corresponding to each successive nestedrotation by the infinitesimal angle ±Δα_(i) or ±Δα_(j′) where themagnitude of the angular sum of the n rotations about each of thei′-axis and the j′-axis is $\frac{\sqrt{2}}{2}{\pi.}$Half of the orbitsphere-cvf is generated during each of Steps One andTwo.

Following Step Two, in order to match the boundary condition that themagnitude of the velocity at any given point on the surface is given byEq. (6), the output half of the orbitsphere-cvf is rotated clockwise byan angle of $\frac{\pi}{4}$about the z-axis. Using Eq. (11) with${\Delta\quad\alpha_{z}},{= \frac{\pi}{4}}$and Δα_(x′)=0 gives the rotation. Then, the one half of theorbitsphere-cvf generated from Step One is superimposed with thecomplementary half obtained from Step Two following its rotation aboutthe z-axis of $\frac{\pi}{4}$to give the basis function to generate Y₀ ⁰(φ,θ), the orbitsphereequation of motion of the electron.

The current pattern of the orbitsphere-cvf generated by the nestedrotations of the orthogonal great circle current loops is a continuousand total coverage of the spherical surface, but it is shown as a visualrepresentation using 6 degree increments of the infinitesimal angularvariable ±Δα_(i′) and ±Δα_(j′) of Eqs. (10) and (11) from theperspective of the z-axis in FIG. 2. In each case, the completeorbitsphere-cvf current pattern corresponds all theorthogonal-great-circle elements which are generated by the rotation ofthe basis-set according to Eqs. (10) and (11) where ±Δα_(i′) and±Δα_(j′) approach zero and the summation of the infinitesimal angularrotations of ±Δα_(i) and ±Δα_(j′) about the successive i′-axes andj′-axes is $\frac{\sqrt{2}}{2}\pi$for each Step. The current pattern gives rise to the phenomenoncorresponding to the spin quantum number. The details of the derivationof the spin function are given in Ref. [3] and Chp. 1 of Ref. [1].

The resultant angular momentum projections of$L_{xy} = {{\frac{\hslash}{4}\quad{and}\quad L_{z}} = \frac{\hslash}{2}}$meet the boundary condition for the unique current having an angularvelocity magnitude at each point on the surface given by Eq. (6) andgive rise to the Stern Gerlach experiment as shown in Ref. [1]. Thefurther constraint that the current density is uniform such that thecharge density is uniform, corresponding to an equipotential, minimumenergy surface is satisfied by using the orbitsphere-cvf as a basiselement to generate Y₀ ⁰(φ,θ) using a convolution operator comprising anautocorrelation-type function as given in Appendix III of Ref. [1]. Theoperator comprises the convolution of each great circle current loop ofthe orbitsphere-cvf designated as the primary orbitsphere-cvf with asecond orbitsphere-cvf designated as the secondary orbitsphere-cvfwherein the convolved secondary elements are matched for orientation,angular momentum, and phase to those of the primary. The resulting exactuniform current distribution obtained from the convolution has the sameangular momentum distribution, resultant, L_(R), and components of$L_{xy} = {{\frac{\hslash}{4}\quad{and}\quad L_{z}} = \frac{\hslash}{2}}$as those of the orbitsphere-cvf used as a primary basis element.1.B. Angular Functions

The time, radial, and angular solutions of the wave equation areseparable. Also based on the radial solution, the angular charge andcurrent-density functions of the electron, A(θ,φ,t), must be a solutionof the wave equation in two dimensions (plus time), $\begin{matrix}{{{\left\lbrack {{\nabla^{2}{- \frac{1}{v^{2}}}}\frac{\partial^{2}}{\partial t^{2}}} \right\rbrack{A\left( {\theta,\phi,t} \right)}} = 0}\begin{matrix}{{{where}\quad{\rho\left( {r,\theta,\phi,t} \right)}} = {{{f(r)}{A\left( {\theta,\phi,t} \right)}} = {\frac{1}{r^{2}}{\delta\left( {r - r_{n}} \right)}}}} \\{{A\left( {\theta,\phi,t} \right)}\quad{and}\quad{A\left( {\theta,\phi,t} \right)}} \\{= {{Y\left( {\theta,\phi} \right)}{k(t)}}}\end{matrix}} & (12) \\\begin{matrix}\left\lbrack {{\frac{1}{r^{2}\sin\quad\theta}\frac{\partial}{\partial\theta}\left( {\sin\quad\theta\frac{\partial}{\partial\theta}} \right)_{r,\phi}} + {\frac{1}{r^{2}\sin^{2}\theta}\left( \frac{\partial^{2}}{\partial\phi^{2}} \right)_{r,0}} - {\frac{1}{v^{2}}\frac{\partial^{2}}{\partial t^{2}}}} \right\rbrack \\{{{A\left( {\theta,\phi,t} \right)} = 0}\quad}\end{matrix} & (13)\end{matrix}$where ν is the linear velocity of the electron. The charge-densityfunctions including the time-function factor are $\begin{matrix}{\ell = 0} & \quad \\{{{\rho\left( {r,\theta,\phi,t} \right)} = {{\frac{e}{8\pi\quad r^{2}}\left\lbrack {\delta\left( {r - r_{n}} \right)} \right\rbrack}\left\lbrack {{Y_{0}^{0}\left( {\theta,\phi} \right)} + {Y_{\ell}^{m}\left( {\theta,\phi} \right)}} \right\rbrack}}{\ell \neq 0}} & (14) \\{{\rho\left( {r,\theta,\phi,t} \right)} = {{\frac{e}{4\pi\quad r^{2}}\left\lbrack {\delta\left( {r - r_{n}} \right)} \right\rbrack}\left\lbrack {{Y_{0}^{0}\left( {\theta,\phi} \right)} + {{Re}\left\{ {{Y_{\ell}^{m}\left( {\theta,\phi} \right)}{\mathbb{e}}^{i\quad\omega_{n}t}} \right\}}} \right\rbrack}} & (15)\end{matrix}$where Y_(l) ^(m)(θ,φ) are the spherical harmonic functions that spinabout the z-axis with angular frequency ω_(n) with Y₀ ⁰(θ,φ) theconstant function. Re{Y_(l) ^(m)(θ,φ)e^(iωj)}=P_(l) ^(m)(cosθ)cos(mφ+{dot over (ω)}_(n)t) where to keep the form of the sphericalharmonic as a traveling wave about the z-axis, {dot over(ω)}_(n)=mω_(n).1.C. Acceleration without Radiation1.C.a. Special Relativistic Correction to the Electron Radius

The relationship between the electron wavelength and its radius is givenby Eq. (4) where λ is the de Broglie wavelength. For each currentdensity element of the spin function, the distance along each greatcircle in the direction of instantaneous motion undergoes lengthcontraction and time dilation. Using a phase matching condition, thewavelengths of the electron and laboratory inertial frames are equated,and the corrected radius is given by $\begin{matrix}\begin{matrix}{r_{n} = {r_{n}\left\lbrack {{\sqrt{1 - \left( \frac{v}{c} \right)^{2}}{\sin\left\lbrack {\frac{\pi}{2}\left( {1 - \left( \frac{v}{c} \right)^{2}} \right)^{3/2}} \right\rbrack}} +} \right.}} \\\left. {\frac{1}{2\pi}{\cos\left\lbrack {\frac{\pi}{2}\left( {1 - \left( \frac{v}{c} \right)^{2}} \right)^{3/2}} \right\rbrack}} \right\rbrack\end{matrix} & (16)\end{matrix}$where the electron velocity is given by Eq. (6). (See Ref. [1] Chp. 1,Special Relativistic Correction to the Ionization Energies section).$\frac{e}{m_{e}}$of the electron, the electron angular momentum of

, and μ_(B) are invariant, but the mass and charge densities increase inthe laboratory frame due to the relativistically contracted electronradius. As ν→c, $\left. {r/r^{\prime}}\rightarrow\frac{1}{2\pi} \right.$and r=λ as shown in FIG. 4.1.C.b. Nonradiation Based on the Spacetime Fourier Transform of theElectron Current

The Fourier transform of the electron charge density function given byEq. (8) is a solution of the three-dimensional wave equation infrequency space (k, ω space) as given in Chp 1, Spacetime FourierTransform of the Electron Function section of Ref. [1]. Then, thecorresponding Fourier transform of the current density functionK(s,Θ,Φ,ω) is given by multiplying by the constant angular frequency.$\begin{matrix}\begin{matrix}{{{K\left( {s,\Theta,\Phi,\omega} \right)} = {4\pi\quad\omega_{n}{\frac{\sin\left( {2s_{n}r_{n}} \right)}{2s_{n}r_{n}} \otimes 2}\pi\quad\sum\limits_{\upsilon = 1}^{\infty}}}\quad} \\{\frac{\left( {- 1} \right)^{\upsilon - 1}\left( {\pi\quad\sin\quad\Theta} \right)^{2{({\upsilon - 1})}}}{{\left( {\upsilon - 1} \right)!}{\left( {\upsilon - 1} \right)!}}} \\{{\frac{{\Gamma\left( \frac{1}{2} \right)}{\Gamma\left( {\upsilon + \frac{1}{2}} \right)}}{\left( {\pi\quad\cos\quad\Theta} \right)^{{2\upsilon} + 1}2^{\upsilon + 1}}\frac{2{\upsilon!}}{\left( {\upsilon - 1} \right)!}{s^{{- 2}\upsilon} \otimes \underset{\upsilon = 1}{\overset{\infty}{2\pi\sum}}}}\quad} \\{\frac{\left( {- 1} \right)^{\upsilon - 1}\left( {\pi\quad\sin\quad\Theta} \right)^{2{({\upsilon - 1})}}}{{\left( {\upsilon - 1} \right)!}{\left( {\upsilon - 1} \right)!}}} \\{\frac{{\Gamma\left( \frac{1}{2} \right)}{\Gamma\left( {\upsilon + \frac{1}{2}} \right)}}{\left( {\pi\quad\cos\quad\Phi} \right)^{{2\upsilon} + 1}2^{\upsilon + 1}}\frac{2{\upsilon!}}{\left( {\upsilon - 1} \right)!}s^{{- 2}\upsilon}} \\{\frac{1}{4\pi}\left\lbrack {{\delta\left( {\omega - \omega_{n}} \right)} + {\delta\left( {\omega + \omega_{n}} \right)}} \right\rbrack}\end{matrix} & (17)\end{matrix}$s_(n)·v_(n)=s_(n)·c=ω_(n) implies r_(n)=λ_(n) which is given by Eq. (16)in the case that k is the lightlike k⁰. In this case, Eq. (17) vanishes.Consequently, spacetime harmonics of$\frac{\omega_{n}}{c} = {{k\quad{or}\quad\frac{\omega_{n}}{c}\sqrt{\frac{ɛ}{ɛ_{o}}}} = k}$for which the Fourier transform of the current-density function isnonzero do not exist. Radiation due to charge motion does not occur inany medium when this boundary condition is met. Nonradiation is alsodetermined directly from the fields based on Maxwell's equations asgiven in Sec. 1.C.c.1.C.c Nonradiation Based on the Electron Electromagnetic Fields and thePoynting Power Vector

A point charge undergoing periodic motion accelerates and as aconsequence radiates according to the Larmor formula: $\begin{matrix}{P = {\frac{1}{4\pi\quad ɛ_{0}}\frac{2e^{2}}{3c^{3}}a^{2}}} & (18)\end{matrix}$where e is the charge, α is its acceleration, ε₀ is the permittivity offree space, and c is the speed of light. Although an accelerated pointparticle radiates, an extended distribution modeled as a superpositionof accelerating charges does not have to radiate [11, 16, 19-21]. InRef. [3] and Appendix I, Chp. 1 of Ref. [1], the electromagnetic farfield is determined from the current distribution in order to obtain thecondition, if it exists, that the electron current distribution mustsatisfy such that the electron does not radiate. The current followsfrom Eqs. (14-15). The currents corresponding to Eq. (14) and first termof Eq. (15) are static. Thus, they are trivially nonradiative. Thecurrent due to the time dependent term of Eq. (15) corresponding to p,d, f, etc. orbitals is $\begin{matrix}\begin{matrix}{J = {\frac{\omega_{n}}{2\pi}\frac{\mathbb{e}}{4\pi\quad r_{n}^{2}}{N\left\lbrack {\delta\left( {r - r_{n}} \right)} \right\rbrack}{Re}{\left\{ {Y_{\ell}^{m}\left( {\theta,\phi} \right)} \right\}\left\lbrack {{u(t)} \times r} \right\rbrack}}} \\{= {\frac{\omega_{n}}{2\pi}\frac{\mathbb{e}}{4\pi\quad r_{n}^{2}}{N^{\prime}\left\lbrack {\delta\left( {r - r_{n}} \right)} \right\rbrack}\left( {{P_{\ell}^{m}\left( {{\cos(\theta)}{\cos\left( {{m\quad\phi} + {\omega_{n}t}} \right)}} \right)}\left\lbrack {u \times r} \right\rbrack} \right.}} \\{= {\frac{\omega_{n}}{2\pi}\frac{\mathbb{e}}{4\pi\quad r_{n}^{2}}{N^{\prime}\left\lbrack {\delta\left( {r - r_{n}} \right)} \right\rbrack}\left( {{P_{\ell}^{m}\left( {{\cos(\theta)}{\cos\left( {{m\quad\phi} + {\omega_{n}t}} \right)}} \right)}\sin\quad\theta\quad\hat{\phi}} \right.}}\end{matrix} & (19)\end{matrix}$where to keep the form of the spherical harmonic as a traveling waveabout the z-axis, {dot over (ω)}_(n)=mω_(n) and N and N′ arenormalization constants. The vectors are defined as $\begin{matrix}{{\hat{\phi} = {\frac{\hat{u} \times \hat{r}}{{\hat{u} \times \hat{r}}} = \frac{\hat{u} \times \hat{r}}{\sin\quad\theta}}};{\hat{u} = {\hat{z} = {{orbital}\quad{axis}}}}} & (20) \\{\hat{\theta} = {\hat{\phi} \times \hat{r}}} & (21)\end{matrix}$“ˆ” denotes the unit vectors ${\hat{u} \equiv \frac{u}{u}},$non-unit vectors are designed in bold, and the current function isnormalized. For the electron source current given by Eq. (19), eachcomprising a multipole of order (l,m) with a time dependence e^(iω) ^(n)^(t), the far-field solutions to Maxwell's equations are given by$\begin{matrix}{{B = {{- \frac{i}{k}}{a_{M}\left( {\ell,m} \right)}{\nabla{\times {g_{\ell}\left( {k\quad r} \right)}X_{\ell,m}}}}}{E = {{a_{M}\left( {\ell,m} \right)}{g_{\ell}\left( {k\quad r} \right)}X_{\ell,m}}}} & (22)\end{matrix}$and the time-averaged power radiated per solid angle$\frac{\mathbb{d}{P\left( {\ell,m} \right)}}{\mathbb{d}\Omega}$is $\begin{matrix}{{\frac{\mathbb{d}{P\left( {\ell,m} \right)}}{\mathbb{d}\Omega} = {\frac{c}{8\pi\quad k^{2}}{{a_{M}\left( {\ell,m} \right)}}^{2}{X_{\ell,m}}^{2}}}{{where}\quad{a_{M}\left( {\ell,m} \right)}\quad{is}}} & (23) \\{{a_{M}\left( {\ell,m} \right)} = {\frac{- {ek}^{2}}{c\sqrt{\ell\left( {\ell + 1} \right)}}\frac{\omega_{n}}{2\pi}{{Nj}_{\ell}\left( {kr}_{n\quad} \right)}{{\Theta sin}({mks})}}} & (24)\end{matrix}$In the case that k is the lightlike k⁰, then k=ω_(n)/c, in Eq. (24), andEqs. (22-23) vanishes fors=vT_(n)=R=r_(n)=λ_(n)  (25)There is no radiation.1.D. Magnetic Field Equations of the Electron

The orbitsphere is a shell of negative charge current comprisingcorrelated charge motion along great circles. For l=0, the orbitspheregives rise to a magnetic moment of 1 Bohr magneton [22]. (The details ofthe derivation of the magnetic parameters including the electron gfactor are given in Ref. [3] and Chp. 1 of Ref. [1].) $\begin{matrix}{\mu_{B} = {\frac{e\quad\hslash}{2m_{e}} = {9.274 \times 10^{- 24}{JT}^{- 1}}}} & (26)\end{matrix}$The magnetic field of the electron shown in FIG. 5 is given by$\begin{matrix}{H = {{\frac{e\quad\hslash}{m_{e}r_{n}^{3}}\left( {{i_{r}\cos\quad\theta} - {i_{0\quad}\sin\quad\theta}} \right)\quad{for}\quad r} < r_{n}}} & (27) \\{H = {{\frac{e\quad\hslash}{2m_{e}r^{3}}\left( {{i_{r}2\cos\quad\theta} + {i_{0\quad}\sin\quad\theta}} \right)\quad{for}\quad r} > r_{n}}} & (28)\end{matrix}$The energy stored in the magnetic field of the electron is$\begin{matrix}{E_{mag} = {\frac{1}{2}\mu_{o}{\int_{0}^{2\pi}{\int_{0}^{\pi}{\int_{0}^{\infty}{H^{2}r^{2}\sin\quad\theta\quad{\mathbb{d}r}\quad{\mathbb{d}\theta}\quad{\mathbb{d}\Phi}}}}}}} & (29) \\{E_{{mag}\quad{total}} = \frac{\pi\quad\mu_{o}{\mathbb{e}}^{2}\hslash^{2}}{m_{e}^{2}r_{1}^{3}}} & (30)\end{matrix}$1.E. Stern-Gerlach Experiment

The Stem-Gerlach experiment implies a magnetic moment of one Bohrmagneton and an associated angular momentum quantum number of ½.Historically, this quantum number is called the spin quantum number,${s\left( {{s = \frac{1}{2}};{m_{s} = {\pm \frac{1}{2}}}} \right)}.$The superposition of the vector projection of the orbitsphere angularmomentum on the z-axis is $\frac{\hslash}{2}$with an orthogonal component of $\frac{\hslash}{4}.$Excitation of a resonant Larmor precession gives rise to

on an axis S that precesses about the z-axis called the spin axis at theLarmor frequency at an angle of $\theta = \frac{\pi}{3}$to give a perpendicular projection of $\begin{matrix}{S_{\bot} = {{\hslash\quad\sin\frac{\pi}{3}} = {{\pm \sqrt{\frac{3}{4}}}\hslash\quad i_{Y_{R}}}}} & (31)\end{matrix}$and a projection onto the axis of the applied magnetic field of$\begin{matrix}{S_{\parallel} = {{\hslash\quad\cos\frac{\pi}{3}} = {{\pm \frac{\hslash}{2}}i_{z}}}} & (32)\end{matrix}$The superposition of the $\frac{\hslash}{2},$z-axis component of the orbitsphere angular momentum and the$\frac{\hslash}{2},$z-axis component of S gives

corresponding to the observed electron magnetic moment of a Bohrmagneton, μ_(B).1.F. Electron g Factor

Conservation of angular momentum of the orbitsphere permits a discretechange of its “kinetic angular momentum” (r×mv) by the applied magneticfield of $\frac{\hslash}{2},$and concomitantly the “potential angular momentum” (r×eA) must change by$- {\frac{\hslash}{2}.}$ $\quad\begin{matrix}\begin{matrix}{{\Delta\quad L} = {\frac{\hslash}{2} - {r \times {\mathbb{e}}\quad A}}} \\{= {\left\lbrack {\frac{\hslash}{2} - \frac{{\mathbb{e}}\quad\phi}{2\pi}} \right\rbrack\hat{z}}}\end{matrix} & \begin{matrix}(33) \\(34)\end{matrix}\end{matrix}$In order that the change of angular momentum, ΔL, equals zero, φ must be${\Phi_{0} = \frac{h}{2{\mathbb{e}}}},$the magnetic flux quantum. The magnetic moment of the electron isparallel or antiparallel to the applied field only. During the spin-fliptransition, power must be conserved. Power flow is governed by thePoynting power theorem, $\begin{matrix}{{\nabla{\cdot \left( {E \times H} \right)}} = {{- {\frac{\partial}{\partial t}\left\lbrack {\frac{1}{2}\mu_{o}{H \cdot H}} \right\rbrack}} - {\frac{\partial}{\partial t}\left\lbrack {\frac{1}{2}ɛ_{o}{E \cdot E}} \right\rbrack} - {J \cdot E}}} & (35)\end{matrix}$Eq. (36) gives the total energy of the flip transition which is the sumof the energy of reorientation of the magnetic moment (1st term), themagnetic energy (2nd term), the electric energy (3rd term), and thedissipated energy of a fluxon treading the orbitsphere (4th term),respectively, $\begin{matrix}{{\Delta\quad E_{mag}^{spin}} = {2\left( {1 + \frac{\alpha}{2\pi} + {\frac{2}{3}{\alpha^{2}\left( \frac{\alpha}{2\pi} \right)}} - {\frac{4}{3}\left( \frac{\alpha}{2\pi} \right)^{2}}} \right)\mu_{B}B}} & (36) \\{{\Delta\quad E_{mag}^{spin}} = {g\quad\mu_{B}B}} & (37)\end{matrix}$where the stored magnetic energy corresponding to the$\frac{\partial}{\partial t}\left\lbrack {\frac{1}{2}\mu_{o}{H \cdot H}} \right\rbrack$term increases, the stored electric energy corresponding to the$\frac{\partial}{\partial t}\left\lbrack {\frac{1}{2}ɛ_{o}{E \cdot E}} \right\rbrack$term increases, and the J·E term is dissipative. The spin-fliptransition can be considered as involving a magnetic moment of g timesthat of a Bohr magneton. The g factor is redesignated the fluxon gfactor as opposed to the anomalous g factor. Using α⁻¹=137.03603(82),the calculated value of $\frac{g}{2}$is 1.001 159 652 137. The experimental value [23] of $\frac{g}{2}$is 1.001 159 652 188(4).1.G. Spin and Orbital Parameters

The total function that describes the spinning motion of each electronorbitsphere is composed of two functions. One function, the spinfunction, is spatially uniform over the orbitsphere, spins with aquantized angular velocity, and gives rise to spin angular momentum. Theother function, the modulation function, can be spatially uniform—inwhich case there is no orbital angular momentum and the magnetic momentof the electron orbitsphere is one Bohr magneton—or not spatiallyuniform—in which case there is orbital angular momentum. The modulationfunction also rotates with a quantized angular velocity.

The spin function of the electron corresponds to the nonradiative n=1,l=0 state of atomic hydrogen which is well known as an s state ororbital. (See FIG. 1 for the charge function and FIG. 2 for the currentfunction.) In cases of orbitals of heavier elements and excited statesof one electron atoms and atoms or ions of heavier elements with the lquantum number not equal to zero and which are not constant as given byEq. (14), the constant spin function is modulated by a time andspherical harmonic function as given by Eq. (15) and shown in FIG. 3.The modulation or traveling charge density wave corresponds to anorbital angular momentum in addition to a spin angular momentum. Thesestates are typically referred to as p, d, f, etc. orbitals. Applicationof Haus's [16] condition also predicts nonradiation for a constant spinfunction modulated by a time and spherically harmonic orbital function.There is acceleration without radiation as also shown in Sec. 1.C.c.(Also see Abbott and Griffiths, Goedecke, and Daboul and Jensen[19-21]). However, in the case that such a state arises as an excitedstate by photon absorption, it is radiative due to a radial dipole termin its current density function since it possesses spacetime FourierTransform components synchronous with waves traveling at the speed oflight [16]. (See Instability of Excited States section of Ref. [1].)

1.G.a Moment of Inertia and Spin and Rotational Energies

The moments of inertia and the rotational energies as a function of thel quantum number for the solutions of the time-dependent electron chargedensity functions (Eqs. (14-15)) given in Sec. 1.B are solved using therigid rotor equation [18]. The details of the derivations of the resultsas well as the demonstration that Eqs. (14-15) with the results giveninfra. are solutions of the wave equation are given in Chp 1, RotationalParameters of the Electron (Angular Momentum, Rotational Energy, Momentof Inertia) section of Ref. [1]. $\begin{matrix}\begin{matrix}{\ell = 0} \\{I_{z} = {I_{spin} = \frac{m_{e}r_{n}^{2}}{2}}}\end{matrix} & (38) \\{L_{z} = {{I\quad\omega\quad i_{z}} = {\pm \frac{\hslash}{2}}}} & (39) \\\begin{matrix}{E_{\quad{rotational}} = E_{\quad{{rotational},\quad{spin}}}} \\{= {\frac{1}{\quad 2}\left\lbrack {\quad{I_{\quad{spin}}\left( \quad\frac{\hslash}{\quad{m_{\quad e}\quad r_{\quad n}^{\quad 2}}} \right)}}^{2} \right\rbrack}} \\{= {\frac{1}{\quad 2}\left\lbrack {\frac{\quad{m_{\quad e}\quad r_{\quad n}^{\quad 2}}}{\quad 2}\left( \quad\frac{\hslash}{\quad{m_{\quad e}\quad r_{\quad n}^{\quad 2}}} \right)^{2}} \right\rbrack}} \\{= {\frac{1}{4}\left\lbrack \frac{\hslash^{2}}{2I_{spin}} \right\rbrack}}\end{matrix} & (40) \\{{T = \frac{\hslash^{2}}{2m_{e}r_{n}^{2}}}{\ell \neq 0}} & (41) \\{I_{orbital} = {{m_{e}{r_{n}^{2}\left\lbrack \frac{\ell\left( {\ell + 1} \right)}{\ell^{2} + {2\ell} + 1} \right\rbrack}^{\frac{1}{2}}} = {m_{e}r_{n}^{2}\sqrt{\frac{\ell}{\ell + 1}}}}} & (42) \\\begin{matrix}{L = {I\quad\omega\quad i_{z}}} \\{= {I_{orbital}\omega\quad i_{z}}} \\{= {m_{e}{r_{n}^{2}\left\lbrack \frac{\ell\left( {\ell + 1} \right)}{\ell^{2} + {2\ell} + 1} \right\rbrack}^{\frac{1}{2}}\omega\quad i_{z}}} \\{= {m_{e}r_{n}^{2}\frac{\hslash}{m_{e}r_{n}^{2}}\sqrt{\frac{\ell}{\ell + 1}}}} \\{= {\hslash\sqrt{\frac{\ell}{\ell + 1}}}}\end{matrix} & (43) \\{L_{z\quad{total}} = {L_{z\quad{spin}} + L_{z\quad{orbital}}}} & (44) \\\begin{matrix}{E_{{rotational}\quad{orbital}} = {\frac{\hslash^{2}}{2I}\left\lbrack \frac{\ell\left( {\ell + 1} \right)}{\ell^{2} + {2\ell} + 1} \right\rbrack}} \\{= {\frac{\hslash^{2}}{2I}\left\lbrack \frac{\ell}{\ell + 1} \right\rbrack}} \\{= {\frac{\hslash^{2}}{2m_{e}r_{n}^{2}}\left\lbrack \frac{\ell}{\ell + 1} \right\rbrack}}\end{matrix} & (45) \\{\left\langle L_{z\quad{orbital}} \right\rangle = 0} & (46) \\{\left\langle E_{{rotational}\quad{orbital}} \right\rangle = 0} & (47)\end{matrix}$The orbital rotational energy arises from a spin function (spin angularmomentum) modulated by a spherical harmonic angular function (orbitalangular momentum). The time-averaged mechanical angular momentum androtational energy associated with the wave-equation solution comprisinga traveling charge-density wave on the orbitsphere is zero as given inEqs. (46) and (47), respectively. Thus, the principal levels aredegenerate except when a magnetic field is applied. In the case of anexcited state, the angular momentum of

is carried by the fields of the trapped photon. The amplitudes thatcouple to external magnetic and electromagnetic fields are given by Eq.(43) and (45), respectively. The rotational energy due to spin is givenby Eq. (40), and the total kinetic energy is given by Eq. (41).1.H. Force Balance Equation

The radius of the nonradiative (n=1) state is solved using theelectromagnetic force equations of Maxwell relating the charge and massdensity functions wherein the angular momentum of the electron is givenby

[1]. The reduced mass arises naturally from an electrodynamicinteraction between the electron and the proton of mass m_(p).$\begin{matrix}{{\frac{m_{e}}{4\pi\quad r_{1}^{2}}\frac{v_{1}^{2}}{r_{1}}} = {{\frac{e}{4\pi\quad r_{1}^{2}}\frac{Ze}{4{\pi ɛ}_{o}r_{1}^{2}}} - {\frac{1}{4\pi\quad r_{1}^{2}}\frac{\hslash}{m_{p}r_{n}^{3}}}}} & (48) \\{r_{1} = \frac{a_{H}}{Z}} & (49)\end{matrix}$where α_(H) is the radius of the hydrogen atom.1.1. Energy Calculations

From Maxwell's equations, the potential energy V, kinetic energy T,electric energy or binding energy E_(ele) are $\begin{matrix}\begin{matrix}{V = \frac{- {Ze}^{2}}{4{\pi ɛ}_{o}r_{1}}} \\{= \frac{{- Z^{2}}e^{2}}{4{\pi ɛ}_{o}a_{H}}} \\{= {{- Z^{2}}\quad X\quad 4.3675\quad X\quad 10^{- 18}J}} \\{= {{- Z^{2}}\quad X\quad 27.2\quad e\quad V}}\end{matrix} & (50) \\{T = {\frac{Z^{2}e^{2}}{8{\pi ɛ}_{o}a_{H}} = {Z^{2}\quad X\quad 13.59\quad e\quad V}}} & (51) \\{\begin{matrix}{T = E_{ele}} \\{= {{- \frac{1}{2}}ɛ_{o}{\int_{\infty}^{r_{1}}{E^{2}\quad{\mathbb{d}v}}}}}\end{matrix}{where}{E = {- \frac{Ze}{4{\pi ɛ}_{o}r^{2}}}}} & (52) \\\begin{matrix}{E_{ele} = {- \frac{{Ze}^{2}}{8{\pi ɛ}_{o}r_{1}}}} \\{= {- \frac{Z^{2}e^{2}}{8{\pi ɛ}_{o}a_{H}}}} \\{= {{- Z^{2}}\quad X\quad 2.1786\quad X\quad 10^{- 18}\quad J}} \\{= {{- Z^{2}}\quad X\quad 13.598\quad{eV}}}\end{matrix} & (53)\end{matrix}$

The calculated Rydberg constant is 10,967,758 m⁻¹; the experimentalRydberg constant is 10,967,758 m⁻¹. For increasing Z, the velocitybecomes a significant fraction of the speed of light; thus, specialrelativistic corrections were included in the calculation of theionization energies of one-electron atoms that are given in TABLE I.TABLE I Relativistically corrected ionization energies for someone-electron atoms. Relative Experimental Difference TheoreticalIonization between One e Ionization Energies Energies Experimental AtomZ γ* ^(a) (eV) ^(b) (eV) ^(c) and Calculated ^(d) H 1 1.000007 13.5983813.59844 0.00000 He⁺ 2 1.000027 54.40941 54.41778 0.00015 Li²⁺ 31.000061 122.43642 122.45429 0.00015 Be³⁺ 4 1.000109 217.68510 217.718650.00015 B⁴⁺ 5 1.000172 340.16367 340.2258 0.00018 C⁵⁺ 6 1.000251489.88324 489.99334 0.00022 N⁶⁺ 7 1.000347 666.85813 667.046 0.00028 O⁷⁺8 1.000461 871.10635 871.4101 0.00035 F⁸⁺ 9 1.000595 1102.650131103.1176 0.00042 Ne⁹⁺ 10 1.000751 1361.51654 1362.1995 0.00050 Na¹⁰⁺ 111.000930 1647.73821 1648.702 0.00058 Mg¹¹⁺ 12 1.001135 1961.354051962.665 0.00067 Al¹²⁺ 13 1.001368 2302.41017 2304.141 0.00075 Si¹³⁺ 141.001631 2670.96078 2673.182 0.00083 P¹⁴⁺ 15 1.001927 3067.069183069.842 0.00090 S¹⁵⁺ 16 1.002260 3490.80890 3494.1892 0.00097 Cl¹⁶⁺ 171.002631 3942.26481 3946.296 0.00102 Ar¹⁷⁺ 18 1.003045 4421.534384426.2296 0.00106 K¹⁸⁺ 19 1.003505 4928.72898 4934.046 0.00108 Ca¹⁹⁺ 201.004014 5463.97524 5469.864 0.00108 Sc²⁰⁺ 21 1.004577 6027.416576033.712 0.00104 Ti²¹⁺ 22 1.005197 6619.21462 6625.82 0.00100 V²²⁺ 231.005879 7239.55091 7246.12 0.00091 Cr²³⁺ 24 1.006626 7888.62855 7894.810.00078 Mn²⁴⁺ 25 1.007444 8566.67392 8571.94 0.00061 Fe²⁵⁺ 26 1.0083389273.93857 9277.69 0.00040 Co²⁶⁺ 27 1.009311 10010.70111 10012.120.00014 Ni²⁷⁺ 28 1.010370 10777.26918 10775.4 −0.00017 Cu²⁸⁺ 29 1.01152011573.98161 11567.617 −0.00055^(a) Eq. (1.250) of Ref. [1] (follows Eqs. (6), (16), and (49)).^(b) Eq. (1.251) of Ref. [1] (Eq. (53) times γ*).^(c) From theoretical calculations, interpolation of H isoelectronic andRydberg series, and experimental data [24-25].^(d) (Experimental-theoretical)/experimental.2. Two Electron Atoms

Two electron atoms may be solved from a central force balance equationwith the nonradiation condition [1]. The centrifugal force,F_(centrifugal), of each electron is given by $\begin{matrix}{F_{centrifugal} = \frac{m_{e}v_{n}^{2}}{r_{n}}} & (54)\end{matrix}$where r_(n) is the radius of electron n which has velocity v_(n). Inorder to be nonradiative, the velocity for every point on theorbitsphere is given by Eq. (6). Now, consider electron 1 initially at$r = {r_{1} = \frac{a_{0}}{Z}}$(the radius of the one-electron atom of charge Z given in the Sec. 1.Hwhere $a_{0} = \frac{4{\pi ɛ}_{0}\hslash^{2}}{e^{2}m_{e}}$and the spin-nuclear interaction corresponding to the electron reducedmass is not used here since the electrons have no field at the nucleusupon pairing) and electron 2 initially at r_(n)=∞. Each electron can betreated as −e charge at the nucleus with$E = \frac{- e}{4{\pi ɛ}_{o}r^{2}}$for r>r_(n) and E=0 for r<r_(n) where r_(n) is the radius of theelectron orbitsphere. The centripetal force is the electric force,F_(ele), between the electron and the nucleus. Thus, the electric forcebetween electron 2 and the nucleus is $\begin{matrix}{F_{{ele}{({{electron}\quad 2})}} = \frac{\left( {Z - 1} \right)e^{2}}{4\quad\pi\quad ɛ_{o}r_{2}^{2}}} & (55)\end{matrix}$where ε_(o) is the permittivity of free-space. The second centripetalforce, F_(mag), on the electron 2 (initially at infinity) from electron1 (at r₁) is the magnetic force. Due to the relative motion of thecharge-density elements of each electron, a radiation reaction forcearises between the two electrons. This force given in Sections 6.6,12.10, and 17.3 of Jackson [26] achieves the condition that the sum ofthe mechanical momentum and electromagnetic momentum is conserved. Themagnetic central force is derived from the Lorentzian force which isrelativistically corrected. The magnetic field of electron 2 at theradius of electron 1 follows from Eq. (1.74b) of Ref. [1] afterMcQuarrie [22]: $\begin{matrix}{B = \frac{\mu_{o}e\quad\hslash}{2\quad m_{e}r_{2}^{3}}} & (56)\end{matrix}$where μ₀ is the permeability of free-space (4π×10⁻¹ N/A²). The motion ateach point of electron 1 in the presence of the magnetic field ofelectron 2 gives rise to a central force which acts at each point ofelectron 2. The Lorentzian force density at each point moving atvelocity v given by Eq. (6) is $\begin{matrix}{F_{mag} = {\frac{e}{4\quad\pi\quad r_{2}^{2}}v \times B}} & (57)\end{matrix}$Substitution of Eq. (6) for v and Eq. (56) for B gives $\begin{matrix}{F_{mag} = {{\frac{1}{4\quad\pi\quad r_{2}^{2}}\left\lbrack \frac{e^{2}\mu_{o}}{2\quad m_{e}r_{1}} \right\rbrack}\frac{\hslash^{2}}{m_{2}r_{2}^{3}}}} & (58)\end{matrix}$The term in brackets can be expressed in terms of the fine structureconstant α. The radius of the electron orbitsphere in the ν=c frame is

_(c), where ν=c corresponds to the magnetic field front propagationvelocity which is the same in all inertial frames, independent of theelectron velocity as shown by the velocity addition formula of specialrelativity [27]. From Eq. (7) and Eqs. (1.144-1.148) of Ref. [1]$\begin{matrix}{\frac{e^{2}\mu_{o}}{2\quad m_{e}r_{1}} = {2\quad\pi\quad\alpha\quad\frac{v}{c}}} & (59)\end{matrix}$where ν=c. Based on the relativistic invariance of the electron'smagnetic moment of a Bohr magneton$\mu_{B} = \frac{e\quad\hslash}{2\quad m_{e}}$as well as its invariant angular momentum of

, it can be shown that the relativistic correction to Eq. (58) is$\frac{1}{Z}$times the reciprocal of Eq. (59). In addition, as given in the SpinAngular Momentum of the Orbitsphere with l=0 section of Ref [1], theapplication of a z-directed magnetic field of electron 2 given by Eq.(1.120) of Ref. [1] to the inner orbitsphere gives rise to a projectionof the angular momentum of electron 1 onto an axis which precesses aboutthe z-axis of $\sqrt{\frac{3}{4}{\hslash.}}$The projection of the force between electron 2 and electron 1 isequivalent to that of the angular momentum onto the axis which precessesabout the z-axis. Thus, Eq. (58) becomes $\begin{matrix}{F_{mag} = {\frac{1}{4\quad\pi\quad r_{2}^{2}}\frac{1}{Z}\frac{\hslash^{2}}{m_{2}r^{3}}\sqrt{s\quad\left( {s + 1} \right)}}} & (60)\end{matrix}$Using Eq. (6), the outward centrifugal force on electron 2 is balancedby the electric force and the magnetic force (on electron 2),$\begin{matrix}{{\frac{m_{e}}{4\quad\pi\quad r_{2}^{2}}\frac{v_{2}^{2}}{r_{2}}} = {{\frac{m_{e}}{4\quad\pi\quad r_{2}^{2}}\frac{\hslash^{2}}{m_{2}r_{2}^{3}}} = {{\frac{e}{4\quad\pi\quad r_{2}^{2}}\frac{\left( {Z - 1} \right)e}{4\quad\pi\quad ɛ_{o}r_{2}^{2}}} + {\frac{1}{4\quad\pi\quad r_{2}^{2}}\frac{\hslash^{2}}{Z\quad m_{e}r_{2}^{3}}\sqrt{s\quad\left( {s + 1} \right)}}}}} & (61)\end{matrix}$which gives the radius of both electrons as $\begin{matrix}{{r_{2} = {r_{1} = {a_{0}\left( {\frac{1}{Z - 1} - \frac{\sqrt{s\left( {s + 1} \right)}}{Z\left( {Z - 1} \right)}} \right)}}};{s = \frac{1}{2}}} & (62)\end{matrix}$(Since the density factor always cancels, it will not be used insubsequent force balance equations).2.A. Ionization Energies Calculated using the Poynting Power Theorem

During ionization, power must be conserved. Power flow is governed bythe Poynting power theorem given by Eq. (35). Energy is superposable;thus, the calculation of the ionization energy is determined as a sum ofthe electric and magnetic contributions. Energy must be supplied toovercome the electric force of the nucleus, and this energy contributionis the negative of the electric work given by Eq. (64). Additionally,the electrons are initially spin paired at r₁=r₂=0.566987 α₀ producingno magnetic fields; whereas, following ionization, the electrons possessmagnetic fields and corresponding energies. For helium, the contributionto the ionization energy is given as the energy stored in the magneticfields of the two electrons at the initial radius where they become spinunpaired. Part of this energy and the corresponding relativistic termcorrespond to the precession of the outer electron about the z-axis dueto the spin angular momentum of the inner electron. These terms are thesame as those of the corresponding terms of the hyperfine structureinterval of muonium as given in the Muonium Hyperfine Structure Intervalsection of Ref [1]. Thus, for helium, which has no electric field beyondr₁ the ionization energy is given by the general formula:$\begin{matrix}{{{{Ionization}\quad{Energy}\quad({He})} = {{- {E({electric})}} + {E\quad({magnetic})\left( {1 - {\frac{1}{2}\left( {\left( {\frac{2}{3}\cos\quad\frac{\pi}{3}} \right)^{2} + \alpha} \right)}} \right)}}}\quad{{where},}} & (63) \\{\quad{{E({electric})} = {- \frac{\left( {Z - 1} \right)e^{2}}{8\quad{\pi ɛ}_{o}r_{1}}}}} & (64) \\{\quad{{E({magnetic})} = {\frac{2\quad\pi\quad\mu_{0}e^{2}\hslash^{2}}{m_{e}^{2}r_{1}^{3}} = \frac{8\quad\pi\quad\mu\quad o_{0}\mu_{B}^{2}}{r_{1}^{3}}}}} & (65)\end{matrix}$Eq. (65) is derived for each of the two electrons as Eq. (1.129) of theMagnetic Parameters of the Electron (Bohr Magneton) section of Ref. [1]with the radius given by Eq. (62).

For 3≦Z, a quantized electric field exists for r>r₁ that gives rise to adissipative term, J·E, of the Poynting Power Vector given by Eq. (35).Thus, the ionization energies are given by $\begin{matrix}{{{Ionization}\quad{Energy}} = {{{- {Electric}}\quad{Energy}} - {\frac{1}{Z}\quad{Magnetic}\quad{Energy}}}} & (66)\end{matrix}$With the substitution of the radius given by Eq. (62) into Eq. (6), thevelocity ν is given by $\begin{matrix}\begin{matrix}{v = \frac{\hslash\quad c}{\sqrt{\left( {\frac{4\quad\pi\quad ɛ_{0}\hslash^{2}}{e^{2}}{c\left( {\frac{1}{Z - 1} - \frac{\sqrt{\frac{3}{4}}}{Z\left( {Z - 1} \right)}} \right)}} \right)^{2} + \hslash^{2}}}} \\{= \frac{\alpha\quad c\quad\left( {Z - 1} \right)}{\sqrt{\left( {1 - \frac{\sqrt{\frac{3}{4}}}{Z}} \right)^{2} + {\alpha^{2}\left( {Z - 1} \right)}^{2}}}}\end{matrix} & (67)\end{matrix}$with Z>1. For increasing Z, the velocity becomes a significant fractionof the speed of light; thus, special relativistic corrections as givenin the Special Relativistic Correction to the Ionization Energiessection of Ref. [1] and Sec. 1.C.a were included in the calculation ofthe ionization energies of two-electron atoms given in TABLE II. Thecalculated ionization energy for helium is 24.58750 eV and theexperimental ionization energy is 24.58741 eV. The agreement in thevalues is within the limit set by experimental error [28].

The solution of the helium atom is further proven to be correct since itis used to solve up through twenty-electron atoms in the Three, Four,Five, Six, Seven, Eight, Nine, Ten, Eleven, Twelve, Thirteen, Fifteen,Sixteen, Seventeen, Eighteen, Nineteen, and Twenty-Electron Atomssection of Ref. [1]. The predictions from general solutions for onethrough twenty-electron atoms are in remarkable agreement with theexperimental values known for 400 atoms and ions. TABLE IIRelativistically corrected ionization energies for some two-electronatoms. Theoretical Experimental Electric Magnetic Ionization ionization2 e r₁ Energy ^(b) Energy ^(c) Velocity Energies Energies Relative AtomZ (α_(o)) ^(a) (eV) (eV) (m/s) ^(d) γ* ^(e) f (eV) g (eV) Error ^(h) He2 0.566987 23.996467 0.590536 3.85845E+06 1.000021 24.58750 24.58741−0.000004 Li⁺ 3 0.35566 76.509 2.543 6.15103E+06 1.00005 75.665 75.64018−0.0003 Be²⁺ 4 0.26116 156.289 6.423 8.37668E+06 1.00010 154.699153.89661 −0.0052 B³⁺ 5 0.20670 263.295 12.956 1.05840E+07 1.00016260.746 259.37521 −0.0053 C⁴⁺ 6 0.17113 397.519 22.828 1.27836E+071.00024 393.809 392.087 −0.0044 N⁵⁺ 7 0.14605 558.958 36.728 1.49794E+071.00033 553.896 552.0718 −0.0033 O⁶⁺ 8 0.12739 747.610 55.3401.71729E+07 1.00044 741.023 739.29 −0.0023 F⁷⁺ 9 0.11297 963.475 79.3521.93649E+07 1.00057 955.211 953.9112 −0.0014 Ne⁸⁺ 10 0.10149 1206.551109.451 2.15560E+07 1.00073 1196.483 1195.8286 −0.0005 Na⁹⁺ 11 0.092131476.840 146.322 2.37465E+07 1.00090 1464.871 1465.121 0.0002 Mg¹⁰⁺ 120.08435 1774.341 190.652 2.59364E+07 1.00110 1760.411 1761.805 0.0008Al¹¹⁺ 13 0.07778 2099.05 243.13 2.81260E+07 1.00133 2083.15 2085.980.0014 Si¹²⁺ 14 0.07216 2450.98 304.44 3.03153E+07 1.00159 2433.132437.63 0.0018 P¹³⁺ 15 0.06730 2830.11 375.26 3.25043E+07 1.001882810.42 2816.91 0.0023 S¹⁴⁺ 16 0.06306 3236.46 456.30 3.46932E+071.00221 3215.09 3223.78 0.0027 Cl¹⁵⁺ 17 0.05932 3670.02 548.223.68819E+07 1.00258 3647.22 3658.521 0.0031 Ar¹⁶⁺ 18 0.05599 4130.79651.72 3.90705E+07 1.00298 4106.91 4120.8857 0.0034 K¹⁷⁺ 19 0.053024618.77 767.49 4.12590E+07 1.00344 4594.25 4610.8 0.0036 Ca¹⁸⁺ 200.05035 5133.96 896.20 4.34475E+07 1.00394 5109.38 5128.8 0.0038 Sc¹⁹⁺21 0.04794 5676.37 1038.56 4.56358E+07 1.00450 5652.43 5674.8 0.0039Ti²⁰⁺ 22 0.04574 6245.98 1195.24 4.78241E+07 1.00511 6223.55 6249 0.0041V²¹⁺ 23 0.04374 6842.81 1366.92 5.00123E+07 1.00578 6822.93 6851.30.0041 Cr²²⁺ 24 0.04191 7466.85 1554.31 5.22005E+07 1.00652 7450.767481.7 0.0041 Mn²³⁺ 25 0.04022 8118.10 1758.08 5.43887E+07 1.007338107.25 8140.6 0.0041 Fe²⁴⁺ 26 0.03867 8796.56 1978.92 5.65768E+071.00821 8792.66 8828 0.0040 Co²⁵⁺ 27 0.03723 9502.23 2217.51 5.87649E+071.00917 9507.25 9544.1 0.0039 Ni²⁶⁺ 28 0.03589 10235.12 2474.556.09529E+07 1.01022 10251.33 10288.8 0.0036 Cu²⁷⁺ 29 0.03465 10995.212750.72 6.31409E+07 1.01136 11025.21 11062.38 0.0034^(a) From Eq. (62).^(b) From Eq. (64).^(c) From Eq. (65).^(d) From Eq. (67).^(e) From Eq. (1.250) of Ref. [1] (follows Eqs. (6), (16), and (49))with the velocity given by Eq. (67).^(f) From Eqs. (63) and (66) with E(electric) of Eq. (64)relativistically corrected by γ* according to Eq.(1.251) of Ref. [1]except that the electron-nuclear electrodynamic relativistic factorcorresponding to the reduced mass of Eqs. (1.213-1.223) was notincluded.^(g) From theoretical calculations for ions Ne⁸⁺ to Cu²⁸⁺ [24-25].^(h) (Experimental-theoretical)/experimental.

The initial central force balance equations with the nonradiationcondition, the initial radii, and the initial energies of the electronsof multi-electron atoms before excitation is given in R. Mills, TheGrand Unified Theory of Classical Quantum Mechanics, January 2005Edition, BlackLight Power, Inc., Cranbury, N.J., (“'05 Mills GUT”) andR. L. Mills, “Exact Classical Quantum Mechanical Solutions forOne-Through Twenty-Electron Atoms”, submitted; posted athttp://www.blacklightpower.com/pdf/technical/Exact%20Classical%20Quantum%20Mechanical%20Solutions%20for%20One-%20Through %20Twenty-Electron%20Atoms %20042204.pdf which areherein incorporated by reference in their entirety.

3. Excited States of Helium

(In this section equation numbers of the form (#.#) correspond to thosegiven in R. Mills, The Grand Unified Theory of Classical QuantumMechanics, January 2005 Edition, BlackLight Power, Inc., Cranbury, N.J.,(“'05 Mills GUT”)).

Bound electrons are described by a charge-density (mass-density)function which is the product of a radial delta function(ƒ(r)=δ(r−r_(n))), two angular functions (spherical harmonic functions),and a time harmonic function. Thus, a bound electron is a dynamic“bubble-like” charge-density function. The two-dimensional sphericalsurface called an electron orbitsphere can exist in a bound state atonly specified distances from the nucleus. More explicitly, theorbitsphere comprises a two-dimensional spherical shell of movingcharge. The current pattern of the orbitsphere that gives rise to thephenomenon corresponding to the spin quantum number comprises aninfinite series of correlated orthogonal great circle current loops. Asgiven in the Orbitsphere Equation of Motion for l=0 section, the currentpattern (shown in FIG. 2) is generated over the surface by twoorthogonal sets of an infinite series of nested rotations of twoorthogonal great circle current loops where the coordinate axes rotatewith the two orthogonal great circles. Each infinitesimal rotation ofthe infinite series is about the new x-axis and new y-axis which resultsfrom the preceding such rotation. For each of the two sets of nestedrotations, the angular sum of the rotations about each rotating x-axisand y-axis totals √{square root over (2)}π radians. The spin function ofthe electron corresponds to the nonradiative n=1, l=0 state which iswell known as an s state or orbital. (See FIG. 1 for the charge functionand FIG. 2 for the current function.) In cases of orbitals of excitedstates with the l quantum number not equal to zero and which are notconstant as given by Eq. (1.64), the constant spin function is modulatedby a time and spherical harmonic function as given by Eq. (1.65) andshown in FIG. 3. The modulation or traveling charge-density wavecorresponds to an orbital angular momentum in addition to a spin angularmomentum. These states are typically referred to as p, d, f, etc.orbitals.

Each orbitsphere is a spherical shell of negative charge (totalcharge=−e) of zero thickness at a distance r_(n) from the nucleus(charge=+Ze). It is well known that the field of a spherical shell ofcharge is zero inside the shell and that of a point charge at the originoutside the shell [29] (See FIG. 1.12 of Ref. [1]). The field of eachelectron can be treated as that corresponding to a −e charge at theorigin with $E = \frac{- e}{4\quad\pi\quad ɛ_{o}r^{2}}$for r>r_(n) and E=0 for r<r_(n) where r_(n) is the radius of theelectron orbitsphere. Thus, as shown in the Two-Electron Atom section ofRef. [1], the central electric fields due to the helium nucleus are$E = \frac{2\quad e}{4\quad\pi\quad ɛ_{o}r^{2}}$and $E = \frac{e}{4\quad\pi\quad ɛ_{o}r^{2}}$for r<r₁ and r₁<r<r₂, respectively. In the ground state of the heliumatom, both electrons are at r₁=r₂=0.567α_(o). When a photon is absorbed,one of the initially indistinguishable electrons called electron 1 movesto a smaller radius, and the other called electron 2 moves to a greaterradius. In the limiting case of the absorption of an ionizing photon,electron 1 moves to the radius of the helium ion, r₁=0.5α_(o), andelectron 2 moves to a continuum radius, r₂=∞. When a photon is absorbedby the ground state helium atom it generates an effective charge,Z_(P-eff), within the second orbitsphere such that the electrons move inopposite radial directions while conserving energy and angular momentum.We can determine Z_(P-eff) of the “trapped photon” electric field byrequiring that the resonance condition is met for photons of discreteenergy, frequency, and wavelength for electron excitation in anelectromagnetic potential energy well.

It is well known that resonator cavities can trap electromagneticradiation of discrete resonant frequencies. The orbitsphere is aresonator cavity which traps single photons of discrete frequencies.Thus, photon absorption occurs as an excitation of a resonator mode. Thefree space photon also comprises a radial Dirac delta function, and theangular momentum of the photon given by$m = {{\int{\frac{1}{8\quad\pi\quad c}{{Re}\left\lbrack {r \times \left( {E \times B^{*}} \right)} \right\rbrack}{\mathbb{d}x^{4}}}} = \hslash}$in the Photon section of Ref. [1] is conserved [30] for the solutionsfor the resonant photons and excited state electron functions as shownfor one-electron atoms in the Excited States of the One-Electron Atom(Quantization) section of Ref. [1]. The correspondence principle holds.That is the change in angular frequency of the electron is equal to theangular frequency of the resonant photon that excites the resonatorcavity mode corresponding to the transition, and the energy is given byPlanck's equation. It can be demonstrated that the resonance conditionbetween these frequencies is to be satisfied in order to have a netchange of the energy field [31].

In general, for a macroscopic multipole with a single m value, acomparison of Eq. (2.33) and Eq. (2.25) shows that the relationshipbetween the angular momentum M_(z), energy U, and angular frequency ω isgiven by Eq. (2.34): $\begin{matrix}{{\frac{\mathbb{d}M_{z}}{\mathbb{d}r} = {\frac{m}{\omega}\frac{\mathbb{d}U}{\mathbb{d}r}}}\quad} & (9.1)\end{matrix}$independent of r where m is an integer. Furthermore, the ratio of thesquare of the angular momentum, M², to the square of the energy, U², fora pure (l, m) multipole follows from Eq. (2.25) and Eqs. (2.31-2.33) asgiven by Eq. (2.35): $\begin{matrix}{\frac{M^{2}}{U^{2}} = \frac{m^{2}}{\omega^{2}}} & (9.2)\end{matrix}$From Jackson [32], the quantum mechanical interpretation is that theradiation from such a multipole of order (l, m) carries off m

units of the z component of angular momentum per photon of energy

ω. However, the photon and the electron can each posses only

of angular momentum which requires that Eqs. (9.1-9.2) correspond to astate of the radiation field containing m photons.

As shown in the Excited States of the One-Electron Atom (Quantization)section of Ref. [1] during excitation the spin, orbital, or totalangular momentum of the orbitsphere can change by zero or ±

. The selection rules for multipole transitions between quantum statesarise from conservation of the photon's multipole moment and angularmomentum of

. In an excited state, the time-averaged mechanical angular momentum androtational energy associated with the traveling charge-density wave onthe orbitsphere is zero (Eq. (1.98)), and the angular momentum of

of the photon that excites the electronic state is carried by the fieldsof the trapped photon. The amplitudes of the rotational energy, momentof inertia, and angular momentum that couple to external magnetic andelectromagnetic fields are given by Eq. (1.95) and (1.96), respectively.Furthermore, the electron charge-density waves are nonradiative due tothe angular motion as shown in the Appendix 1: Nonradiation Based on theElectromagnetic Fields and the Poynting Power Vector section of Ref.[1]. But, excited states are radiative due to a radial dipole thatarises from the presence of the trapped photon as shown in theInstability of Excited States section of Ref. [1] corresponding to m=1in Eqs. (9.1-9.2).

Then, as shown in the Excited States of the One-Electron Atom(Quantization) section and the Derivation of the Rotational Parametersof the Electron section of Ref. [1], the total number of multipoles,N_(l,s), of an energy level corresponding to a principal quantum numbern where each multipole corresponds to an l and ml quantum number is$\begin{matrix}{N_{l,s} = {{\sum\limits_{t = 0}^{n - 1}\quad{\sum\limits_{m_{l} = {- l}}^{+ l}\quad 1}} = {{{\sum\limits_{l = 0}^{n - 1}\quad{2l}} + 1} = {\left( {l + 1} \right)^{2} = {{l^{2} + {2l} + 1} = n^{2}}}}}} & (9.3)\end{matrix}$Any given state may be due to a direct transition or due to the sum oftransitions between all intermediate states wherein the multiplicity ofpossible multipoles increases with higher states. Then, therelationships between the parameters of Eqs. (9.1) and (9.2) due totransitions of quantized angular momentum

, energy

ω, and radiative via a radial dipole are given by substitution of m=1and normalization of the energy U by the total number of degeneratemultipoles, n². This requires that the photon's electric fieldsuperposes that of the nucleus for r₁<r<r₂ such that the radial electricfield has a magnitude proportional to e/n at the electron 2 where n=2,3, 4, . . . for excited states such that U is decreased by the factor of1/n².

Energy is conserved between the electric and magnetic energies of thehelium atom as shown by Eq. (7.26). The helium atom and the “trappedphoton” corresponding to a transition to a resonant excited state haveneutral charge and obey Maxwell's equations. Since charge isrelativistically invariant, the energies in the electric and magneticfields of the electrons of the helium atom must be conserved as photonsare emitted or absorbed. The corresponding forces are determined fromthe requirement that the radial excited-state electric field has amagnitude proportional to e/n at electron 2.

The “trapped photon” is a “standing electromagnetic wave” which actuallyis a traveling wave that propagates on the surface around the z-axis,and its source current is only at the orbitsphere. The time-functionfactor, k(t), for the “standing wave” is identical to the time-functionfactor of the orbitsphere in order to satisfy the boundary (phase)condition at the orbitsphere surface. Thus, the angular frequency of the“trapped photon” has to be identical to the angular frequency of theelectron orbitsphere, ω_(n), given by Eq. (1.55). Furthermore, the phasecondition requires that the angular functions of the “trapped photon”have to be identical to the spherical harmonic angular functions of theelectron orbitsphere. Combining k(t) with the φ-function factor of thespherical harmonic gives e^(i(mφ−ω) ^(n) ^(t)) for both the electron andthe “trapped photon” function.

The photon “standing wave” in an excited electronic state is a solutionof Laplace's equation in spherical coordinates with source currentsgiven by Eq. (2.11) “glued” to the electron and phase-locked to theelectron current density wave that travel on the surface with a radialelectric field. As given in the Excited States of the One-Electron Atom(Quantization) section of Ref. [1], the photon field is purely radialsince the field is traveling azimuthally at the speed of light eventhough the spherical harmonic function has a velocity less than lightspeed given by Eq. (1.56). The photon field does not change the natureof the electrostatic field of the nucleus or its energy except at theposition of the electron. The photon “standing wave” function comprisesa radial Dirac delta function that “samples” the Laplace equationsolution only at the position infinitesimally inside of the electroncurrent-density function and superimposes with the proton field to givea field of radial magnitude corresponding to a charge of e/n where n,=2,3, 4, . . . .

The electric field of the nucleus for r₁<r<r₂ is $\begin{matrix}{E_{nucleus} = \frac{\mathbb{e}}{4{\pi ɛ}_{o}r^{2}}} & (9.4)\end{matrix}$From Eq. (2.15), the equation of the electric field of the “trappedphoton” for r=r₂ where r₂ is the radius of electron 2, is$\begin{matrix}{{E_{r_{{{photon}\quad n},l,{m|_{y = r_{2}}}}} = {{\frac{\mathbb{e}}{4{\pi ɛ}_{o}r_{2}^{2}}\left\lbrack {{- 1} + {\frac{1}{n}\left\lbrack {{Y_{0}^{0}\left( {\theta,\phi} \right)} + {{Re}\left\{ {{Y_{l}^{m}\left( {\theta,\phi} \right)}{\mathbb{e}}^{{\mathbb{i}\omega}_{n}l}} \right\}}} \right\rbrack}} \right\rbrack}{\delta\left( {r - r_{n}} \right)}}}{\omega_{n} = {{0\quad{for}\quad m} = 0}}} & (9.5)\end{matrix}$The total central field for r=r₂ is given by the sum of the electricfield of the nucleus and the electric field of the “trapped photon”.E _(total) =E _(nucleus) +E _(photon)  (9.6)Substitution of Eqs. (9.4) and (9.5) into Eq. (9.6) gives for r₁<r<r₂,$\begin{matrix}{\begin{matrix}{E_{r_{total}} = \begin{matrix}{\frac{\mathbb{e}}{4{\pi ɛ}_{o}r_{1}^{2}} +} \\{\frac{\mathbb{e}}{4{\pi ɛ}_{o}r_{2}^{2}}\left\lbrack {{- 1} + {\frac{1}{n}\left\lbrack {{Y_{0}^{0}\left( {\theta,\phi} \right)} + {{Re}\left\{ {{Y_{l}^{m}\left( {\theta,\phi} \right)}{\mathbb{e}}^{{\mathbb{i}\omega}_{n}l}} \right\}}} \right\rbrack}} \right\rbrack} \\{\delta\left( {r - r_{n}} \right)}\end{matrix}} \\{= {\frac{1}{n}{\frac{\mathbb{e}}{4{\pi ɛ}_{o}r_{2}^{2}}\left\lbrack {{Y_{0}^{0}\left( {\theta,\phi} \right)} + {{Re}\left\{ {{Y_{l}^{m}\left( {\theta,\phi} \right)}{\mathbb{e}}^{{\mathbb{i}\omega}_{n}l}} \right\}}} \right\rbrack}{\delta\left( {r - r_{n}} \right)}}}\end{matrix}{\omega_{n} = {{0\quad{for}\quad m} = 0}}} & (9.7)\end{matrix}$For r=r₂ and m=0, the total radial electric field is $\begin{matrix}{E_{r_{total}} = {\frac{1}{n}\frac{\mathbb{e}}{4{\pi ɛ}_{o}r^{2}}}} & (9.8)\end{matrix}$The result is equivalent to Eq. (2.17) of the Excited States of theOne-Electron Atom (Quantization) section of Ref. [1].

In contrast to short comings of quantum-mechanical equations, withclassical quantum mechanics (CQM), all excited states of the helium atomcan be exactly solved in closed form. The radii of electron 2 aredetermined from the force balance of the electric, magnetic, andcentrifugal forces that corresponds to the minimum of energy of thesystem. The excited-state energies are then given by the electricenergies at these radii. All singlet and triplet states with l=0 or l≠0are solved exactly except for small terms corresponding to themagnetostatic energies in the magnetic fields of excited-stateelectrons, spin-nuclear interactions, and the very small term due tospin-orbital coupling. In the case of spin-nuclear interactions, α_(He)which includes the reduced electron mass according to Eqs. (1.221-1.224)was used rather than α₀ as a partial correction, and a table of thespin-orbital energies was calculated for l=1 to compare to the effect ofdifferent l quantum numbers. For over 100 states, the agreement betweenthe predicted and experimental results are remarkable.

3.A Singlet Excited States with l=0 (1s²→1s¹(ns)¹)

With l=0, the electron source current in the excited state is a constantfunction given by Eq. (1.64) that spins as a globe about the z-axis:$\begin{matrix}{{\rho\left( {r,\theta,\phi,t} \right)} = {{\frac{\mathbb{e}}{8\pi\quad r^{2}}\left\lbrack {\delta\left( {r - r_{n}} \right)} \right\rbrack}\left\lbrack {{Y_{0}^{0}\left( {\theta,\phi} \right)} + {Y_{l}^{m}\left( {\theta,\phi} \right)}} \right\rbrack}} & (9.9)\end{matrix}$As given in the Derivation of the Magnetic Field section in Chapter Oneof Ref. [1] and by Eq. (12.342), the current is a function of sin θwhich gives rise to a correction of ⅔ to the field given by Eq. (7.4)and, correspondingly, the magnetic force of two-electron atoms given byEq. (7.15). The balance between the centrifugal and electric andmagnetic forces is given by the Eq. (7.18): $\begin{matrix}{\frac{m_{e}v^{2}}{r_{2}} = {\frac{\hslash^{2}}{m_{e}r_{2}^{3}} = {{\frac{1}{n}\frac{{\mathbb{e}}^{2}}{4{\pi ɛ}_{o}r_{2}^{2}}} + {\frac{2}{3}\frac{1}{n}\frac{\hslash^{2}}{2m_{e}r_{2}^{3}}\sqrt{s\left( {s + 1} \right)}}}}} & (9.10)\end{matrix}$with the exceptions that the electric and magnetic forces are reduced bya factor of $\frac{1}{n}$since the corresponding charge from Eq. (9.8) is $\frac{\mathbb{e}}{n}$and the magnetic force is further corrected by the factor of ⅔. With${s = \frac{1}{2}},$ $\begin{matrix}{{r_{2} = {\left\lbrack {n - \frac{\sqrt{\frac{3}{4}}}{3}} \right\rbrack\alpha_{He}}}{{n = 2},3,4,\ldots}} & (9.11)\end{matrix}$

The excited-state energy is the energy stored in the electric field,E_(ele), given by Eqs. (1.232), (1.233), and (10.102) which is theenergy of electron 2 relative to the ionized electron at rest havingzero energy: $\begin{matrix}{E_{ele} = {{- \frac{1}{n}}\frac{{\mathbb{e}}^{2}}{8{\pi ɛ}_{o}r_{2}}}} & (9.12)\end{matrix}$where r₂ is given by Eq. (9.11) and from Eq. (9.8), Z=1/n in Eq.(1.233). The energies of the various singlet excited states of heliumwith l=0 appear in TABLE III.

As shown in the Special Relativistic Correction to the IonizationEnergies section of Ref. [1] and Sec. 1.C.a the electron possesses aninvariant charge-to-mass ratio $\left( \frac{\mathbb{e}}{m_{e}} \right)$angular momentum of

, and magnetic moment of a Bohr magneton (μ_(B)). This invariancefeature provides for the stability of multielectron atoms as shown inthe Two-Electron Atom section of Ref. [1] and the Three, Four, Five,Six, Seven, Eight, Nine, Ten, Eleven, Twelve, Thirteen, Fourteen,Fifteen, Sixteen, Seventeen, Eighteen, Nineteen, and Twenty-ElectronAtoms section of Ref. [1]. This feature also permits the existence ofexcited states wherein electrons magnetically interact. The electron'smotion corresponds to a current which gives rise to a magnetic fieldwith a field strength that is inversely proportional to its radius cubedas given in Eq. (9.10) wherein the magnetic field is a relativisticeffect of the electric field as shown by Jackson [33]. Since the forceson electron 2 due to the nucleus and electron 1 (Eq. (9.10)) areradial/central, invariant of r₁, and independent of r₁ with thecondition that r₁<r₂, r₂ can be determined without knowledge of r₁. But,once r₂ is determined, r₁ can be solved using the equal and oppositemagnetic force of electron 2 on electron 1 and the central Coulombicforce corresponding to the nuclear charge of 2e. Using Eq. (9.10), theforce balance between the centrifugal and electric and magnetic forcesis $\begin{matrix}{{\frac{m_{e}v^{2}}{r_{1}} = {\frac{\hslash^{2}}{m_{e}r_{1}^{3}} = {\frac{2{\mathbb{e}}^{2}}{4{\pi ɛ}_{o}r_{1}^{2}} - {\frac{1}{3n}\frac{\hslash^{2}}{m_{e}r_{2}^{3}}\sqrt{s\left( {s + 1} \right)}}}}}{{{{With}\quad s} = \frac{1}{2}},}} & (9.13) \\{{{r_{1}^{3} - {\left( {\frac{12n}{\sqrt{3}}r_{2}^{3}} \right)r_{1}} + {\frac{6n}{\sqrt{3}}r_{2}^{3}}} = 0}{{n = 2},3,4,\ldots}} & (9.14)\end{matrix}$where r₂ is given by Eq. (9.11) and r₁ and r₂ are in units of α_(He). Toobtain the solution of cubic Eq. (9.14), let $\begin{matrix}{g = {\frac{6n}{\sqrt{3}}r_{2}^{3}}} & (9.15)\end{matrix}$Then, Eq. (9.14) becomesr ₁ ³−2gr ₁ +g=0 n=2, 3, 4, . . .   (9.16)and the roots are $\begin{matrix}{r_{11} = {A + B}} & (9.17) \\{r_{12} = {{- \frac{A + B}{2}} + {\frac{A - B}{2}\sqrt{- 3}}}} & (9.18) \\{{r_{13} = {{- \frac{A + B}{2}} - {\frac{A - B}{2}\sqrt{- 3}}}}{where}} & (9.19) \\{{A = {\sqrt[3]{{- \frac{g}{2}} + \sqrt{\frac{g^{2}}{4} - \frac{8g^{3}}{27}}} = {\sqrt[3]{\frac{g}{2}}\sqrt[3]{z}}}}{and}} & (9.20) \\{B = {\sqrt[3]{{- \frac{g}{2}} - \sqrt{\frac{g^{2}}{4} - \frac{8g^{3}}{27}}} = {\sqrt[3]{\frac{g}{2}}\sqrt[3]{\overset{\_}{z}}}}} & (9.21)\end{matrix}$The complex number z is defined by $\begin{matrix}{z = {{{- 1} + \sqrt[i]{{\frac{32}{27}g} - 1}} = {{re}^{i0} = {r\left( {{\cos\quad\theta} + {i\quad\sin\quad\theta}} \right)}}}} & (9.22)\end{matrix}$where the modulus, r, and argument, θ, are $\begin{matrix}{{r = \sqrt{\frac{32}{27}g}}{and}} & (9.23) \\{\theta = {\frac{\pi}{2} + {\sin^{- 1}\left( {1/r} \right)}}} & (9.24)\end{matrix}$respectively. The cube roots are $\begin{matrix}{\sqrt[3]{z} = {{\sqrt[3]{r}e^{i\quad{\theta/3}}} = {\sqrt[3]{r}\left( {{\cos\frac{\theta}{3}} + {i\quad\sin\frac{\theta}{3}}} \right)}}} & (9.25) \\{{\sqrt[3]{\overset{\_}{z}} = {{\sqrt[3]{r}e^{{- i}\quad{\theta/3}}} = {\sqrt[3]{r}\left( {{\cos\frac{\theta}{3}} - {i\quad\sin\frac{\theta}{3}}} \right)}}}{{So},}} & (9.26) \\{{A = {\sqrt[3]{\frac{g}{2}r}\left( {{\cos\frac{\theta}{3}} + {i\quad\sin\frac{\theta}{3}}} \right)}}{and}} & (9.27) \\{B = {\sqrt[3]{\frac{g}{2}r}\left( {{\cos\frac{\theta}{3}} - {i\quad\sin\frac{\theta}{3}}} \right)}} & (9.28)\end{matrix}$

The real and physical root is $\begin{matrix}{r_{1} = {r_{13} = {{- \sqrt{\frac{2}{3}g}}\left( {{\cos\frac{\theta}{3}} - {\sqrt{3}\sin\frac{\theta}{3}}} \right)}}} & (9.29)\end{matrix}$ TABLE III Calculated and experimental energies of He Isinglet excited states with l = 0 (1s² → 1s¹(ns)¹). E_(ele) CQM NIST HeI Energy He I Energy Difference Relative r₁ r₂ Term Levels ^(c) Levels^(d) CQM − NIST Difference ^(e) n (α_(He)) ^(a) (α_(He)) ^(b) Symbol(eV) (eV) (eV) (CQM − NIST) 2 0.501820 1.71132 1s2s ¹S −3.97465 −3.97161−0.00304 0.00077 3 0.500302 2.71132 1s3s ¹S −1.67247 −1.66707 −0.005400.00324 4 0.500088 3.71132 1s4s ¹S −0.91637 −0.91381 −0.00256 0.00281 50.500035 4.71132 1s5s ¹S −0.57750 −0.57617 −0.00133 0.00230 6 0.5000165.71132 1s6s ¹S −0.39698 −0.39622 −0.00076 0.00193 7 0.500009 6.711321s7s ¹S −0.28957 −0.2891 −0.00047 0.00163 8 0.500005 7.71132 1s8s ¹S−0.22052 −0.2202 −0.00032 0.00144 9 0.500003 8.71132 1s9s ¹S −0.17351−0.1733 −0.00021 0.00124 10 0.500002 9.71132 1s10s ¹S −0.14008 −0.13992−0.00016 0.00116 11 0.500001 10.71132 1s11s ¹S −0.11546 −0.11534−0.00012 0.00103 Avg. −0.00144 0.00175^(a) Radius of the inner electron 1 from Eq. (9.29).^(b) Radius of the outer electron 2 from Eq. (9.11).^(e) Classical quantum mechanical (CQM) calculated energy levels givenby the electric energy (Eq. (9.12)).^(d) Experimental NIST levels [34] with the ionization potential definedas zero.^(e) (Theoretical-Experimental)/Experimental.3.B Triplet Excited States with l=(1s²→1s¹(ns)^(i))

For l=0, time-independent charge-density waves corresponding to thesource currents travel on the surface of the orbitsphere of electron 2about the z-axis at the angular frequency given by Eq. (1.55). In thecase of singlet states, the current due to spin of electron 1 andelectron 2 rotate in opposite directions; whereas, for triplet states,the relative motion of the spin currents is in the same direction. Inthe triplet state, the electrons are spin-unpaired, but due to thesuperposition of the excited state source currents and the currentcorresponding to the spin-unpairing transition to create the tripletstate, the spin-spin force is paramagnetic. The angular momentumcorresponding to the excited states is

and the angular momentum change corresponding to the spin-flip or 180°rotation of the Larmor precession is also

as given in the Magnetic Parameters of the Electron (Bohr Magneton)section of Ref. [1]. The maximum projection of the angular momentum of aconstant function onto a defined axis (Eq. (1.74a)) is $\begin{matrix}{S_{\bot} = {{\hslash\quad\sin\quad\frac{\pi}{3}} = {{\pm \sqrt{\frac{3}{4}}}\hslash\quad i_{Y_{k}}}}} & (9.30)\end{matrix}$Following the derivation for Eq. (7.15) using Eq. (9.30) and a magneticmoment of 2μ_(B) corresponding to a total angular momentum of theexcited triplet state of 2

, the spin-spin force for electron 2 is twice that of the singletstates: $\begin{matrix}{{\frac{m_{e}y^{2}}{r_{2}} = {\frac{\hslash^{2}}{m_{e}r_{2}^{3}} = {{\frac{1}{n}\frac{e^{2}}{4\pi\quad ɛ_{o}r_{2}^{2}}} + {2\quad\frac{2}{3}\frac{1}{n}\frac{\hslash^{2}}{2m_{e}r_{2}^{3}}\sqrt{s\left( {s + 1} \right)}}}}}{{{{With}\quad s} = \frac{1}{2}},}} & (9.31) \\{{r_{2} = {{\left\lbrack {n - \frac{2\sqrt{\frac{3}{4}}}{3}} \right\rbrack a_{He}\quad n} = 2}},3,4,\ldots} & (9.32)\end{matrix}$The excited-state energy is the energy stored in the electric field,E_(ele), given by Eq. (9.12) where r₂ is given by Eq. (9.32). Theenergies of the various triplet excited states of helium with l=0 appearin TABLE IV.

Using r₂ (Eq. (9.32), r₁ can be solved using the equal and oppositemagnetic force of electron 2 on electron 1 and the central Coulombicforce corresponding to the nuclear charge of 2e. Using Eq. (9.31), theforce balance between the centrifugal and electric and magnetic forcesis $\begin{matrix}{{\frac{m_{e}y^{2}}{r_{1}} = {\frac{\hslash^{2}}{m_{e}r_{1}^{3}} = {\frac{2e^{2}}{4\pi\quad ɛ_{o}r_{1}^{3}} - {\frac{2}{3n}\frac{\hslash^{2}}{m_{e}r_{2}^{3}}\sqrt{s\left( {s + 1} \right)}}}}}{{{{With}\quad s} = \frac{1}{2}},}} & (9.33) \\{{{r_{1}^{3} - {\left( {\frac{6n}{\sqrt{3}}r_{2}^{3}} \right)r_{1}} + {\frac{3n}{\sqrt{3}}r_{2}^{3}}} = 0}\quad{{n = 2},3,4,\ldots}} & (9.34)\end{matrix}$where r₂ is given by Eq. (9.32) and r₁ and r₂ are in units of α_(He). Toobtain the solution of cubic Eq. (9.34), let $\begin{matrix}{g = {\frac{3n}{\sqrt{3}}r_{2}^{3}}} & (9.35)\end{matrix}$Then, Eq. (9.34) becomesr ₁ ³−2gr ₁ +g=0 n=2, 3, 4, . . .   (9.36)

Using Eqs. (9.16-9.29), the real and physical root is $\begin{matrix}{r_{1} = {r_{13} = {{- \sqrt{\frac{2}{3}}}{g\left( {{\cos\quad\frac{\theta}{3}} - {\sqrt{3}\sin\quad\frac{\theta}{3}}} \right)}}}} & (9.37)\end{matrix}$ TABLE IV Calculated and experimental energies of He Itriplet excited states with l = 0 (1s² → 1s¹(ns)¹). E_(ele) CQM NIST HeI Energy He I Energy Difference Relative r₁ r₂ Term Levels ^(c) Levels^(d) CQM − NIST Difference ^(e) n (α_(He)) ^(a) (α_(He)) ^(b) Symbol(eV) (eV) (eV) (CQM − NIST) 2 0.506514 1.42265 1s2s ³S −4.78116 −4.76777−0.01339 0.00281 3 0.500850 2.42265 1s3s ³S −1.87176 −1.86892 −0.002840.00152 4 0.500225 3.42265 1s4s ³S −0.99366 −0.99342 −0.00024 0.00024 50.500083 4.42265 1s5s ³S −0.61519 −0.61541 0.00022 −0.00036 6 0.5000385.42265 1s6s ³S −0.41812 −0.41838 0.00026 −0.00063 7 0.500019 6.422651s7s ³S −0.30259 −0.30282 0.00023 −0.00077 8 0.500011 7.42265 1s8s ³S−0.22909 −0.22928 0.00019 −0.00081 9 0.500007 8.42265 1s9s ³S −0.17946−0.17961 0.00015 −0.00083 10 0.500004 9.42265 1s10s ³S −0.14437 −0.14450.00013 −0.00087 11 0.500003 10.42265 1s11s ³S −0.11866 −0.11876 0.00010−0.00087 Avg. −0.00152 −0.00006^(a) Radius of the inner electron 1 from Eq. (9.37).^(b) Radius of the outer electron 2 from Eq. (9.32).^(e) Classical quantum mechanical (CQM) calculated energy levels givenby the electric energy (Eq. (9.12)).^(d) Experimental NIST levels [34] with the ionization potential definedas zero.^(e) (Theoretical-Experimental)/Experimental.3.C Singlet Excited States with l≠0

With l≠0, the electron source current in the excited state is the sum ofconstant and time-dependent functions where the latter, given by Eq.(1.65), travels about the z-axis. The current due to the time dependentterm of Eq. (1.65) corresponding to p, d, f, etc. orbitals is$\begin{matrix}\begin{matrix}{J = {\frac{\omega_{n}}{2\pi}\frac{e}{4\pi_{n}^{2}}{N\left\lbrack {\delta\left( {r - r_{n}} \right)} \right\rbrack}{Re}{\left\{ {Y_{\ell}^{m}\left( {\theta,\phi} \right)} \right\}\left\lbrack {{u(t)} \times r} \right\rbrack}}} \\{= {\frac{\omega_{n}}{2\pi}\frac{e}{4\pi_{n}^{2}}{N^{\prime}\left\lbrack {\delta\left( {r - r_{n}} \right)} \right\rbrack}{\left( {{P_{\ell}^{m}\left( {\cos\quad\theta} \right)}{\cos\left( {{m\quad\phi} + {\omega_{n}^{\prime}t}} \right)}} \right)\left\lbrack {u \times r} \right\rbrack}}} \\{= {\frac{\omega_{n}}{2\pi}\frac{e}{4\pi_{n}^{2}}{N^{\prime}\left\lbrack {\delta\left( {r - r_{n}} \right)} \right\rbrack}\left( {{P_{\ell}^{m}\left( {\cos\quad\theta} \right)}{\cos\left( {{m\quad\phi} + {\omega_{n}^{\prime}t}} \right)}} \right)\sin\quad\theta\quad\hat{\phi}}}\end{matrix} & (9.38)\end{matrix}$where to keep the form of the spherical harmonic as a traveling waveabout the z-axis, {dot over (ω)}_(n)=mω_(n) and N and N′ arenormalization constants. The vectors are defined as $\begin{matrix}{{\hat{\phi} = {\frac{\hat{u} \times \hat{r}}{{\hat{u} \times \hat{r}}} = \frac{\hat{u} \times \hat{r}}{\sin\quad\theta}}};{\hat{u} = {\hat{z} = {{orbital}\quad{axis}}}}} & (9.39) \\{\hat{\theta} = {\hat{\phi} \times \hat{r}}} & (9.40)\end{matrix}$“ˆ” denotes the unit vectors ${\hat{u} \equiv \frac{u}{u}},$non-unit vectors are designed in bold, and the current function isnormalized.

Jackson [35] gives the general multipole field solution to Maxwell'sequations in a source-free region of empty space with the assumption ofa time dependence e^(iω) ^(n) ^(t): $\begin{matrix}{{B = {\sum\limits_{\ell,m}\left\lbrack {{{a_{E}\left( {\ell,m} \right)}{f_{\ell}({kr})}X_{\ell,m}} - {\frac{\mathbb{i}}{k}{a_{M}\left( {\ell,m} \right)}{\nabla{\times {g_{\ell}({kr})}X_{\ell,m}}}}} \right\rbrack}}{E = {\sum\limits_{\ell,m}\left\lbrack {{\frac{\mathbb{i}}{k}{a_{E}\left( {\ell,m} \right)}{\nabla{\times {f_{\ell}({kr})}X_{\ell,m}}}} + {{a_{M}\left( {\ell,m} \right)}{g_{\ell}({kr})}X_{\ell,m}}} \right\rbrack}}} & (9.41)\end{matrix}$where the cgs units used by Jackson are retained in this section. Theradial functions f_(l)(kr) and g_(l)(kr) are of the form:·g _(l)(kr)=A _(t) ⁽¹⁾ h _(l) ⁽¹⁾ +A _(l) ⁽²⁾ h _(l) ⁽²⁾  (9.42)X_(l,m) is the vector spherical harmonic defined by $\begin{matrix}{{{X_{\ell,m}\left( {\theta,\phi} \right)} = {\frac{1}{\sqrt{\ell\left( {\ell + 1} \right)}}L\quad{Y_{\ell,m}\left( {\theta,\phi} \right)}}}{where}} & (9.43) \\{L = {\frac{1}{\mathbb{i}}\left( {r \times \nabla} \right)}} & (9.43)\end{matrix}$The coefficients α_(E)(l,m) and α_(M)(l,m) of Eq. (9.41) specify theamounts of electric (l,m) multipole and magnetic (l,m) multipole fields,and are determined by sources and boundary conditions as are therelative proportions in Eq. (9.42). Jackson gives the result of theelectric and magnetic coefficients from the sources as $\begin{matrix}{{\alpha_{E}\left( {\ell,m} \right)} = {\frac{4\pi\quad k^{2}}{i\sqrt{\ell\left( {\ell + 1} \right)}}\quad{\int{Y_{\ell}^{m^{\prime}}\left\{ {{\rho{\frac{\delta}{\delta\quad r}\left\lbrack {{rj}_{\ell}({kr})} \right\rbrack}} + {\frac{ik}{c}\left( {r \cdot J} \right){j_{\ell}({kr})}} - {{ik}{\nabla{\cdot \left( {r \times M} \right)}}{j_{\ell}({kr})}}} \right\}{\mathbb{d}^{3}x}}}}} & (9.45) \\{{\alpha_{M}\left( {\ell,m} \right)} = {\frac{{- 4}\pi\quad k^{2}}{\sqrt{\ell\left( {\ell + 1} \right)}}{\int{{j_{\ell}({kr})}y_{\ell}^{m^{\prime}}{L \cdot \left( {\frac{J}{c} + {\nabla{\times M}}} \right)}{\mathbb{d}^{3}x}}}}} & (9.46)\end{matrix}$respectively, where the distribution of charge ρ(x,t), current J(x,t),and intrinsic magnetization M(x,t) are harmonically varying sources:ρ(x)e^(−ω) ^(n) ^(t), J(x)e^(−ω) ^(n) ^(t), and M(x)e^(−ωj). From Eq.(9.38), the charge and intrinsic magnetization terms are zero. Since thesource dimensions are very small compared to a wavelength (kr_(max)<<1),the small argument limit can be used to give the magnetic multipolecoefficient α_(M)(l,m) as $\begin{matrix}\begin{matrix}{{\alpha_{M}\quad\left( {\ell,m} \right)} = {\frac{{- 4}\pi\quad k^{l + 2}}{\left( {{2\ell} + 1} \right)!!}\left( \frac{\ell + 1}{\ell} \right)^{1/2}}} \\{\left( {M_{\ell\quad m} + M_{\ell\quad m}^{\prime}} \right) = \frac{{- 4}\pi\quad k^{l + 2}}{\frac{\left( {{2\ell} + 1} \right)!}{2^{n}{n!}}}}\end{matrix} & (9.47)\end{matrix}$where the magnetic multipole moments are $\begin{matrix}\begin{matrix}{M_{\ell\quad m} = {{- \frac{1}{\ell + 1}}{\int{r^{\ell}Y_{\ell\quad m}^{*}{\nabla{\cdot \left( \frac{r \times J}{c} \right)}}{\mathbb{d}^{3}x}}}}} \\{M_{\ell\quad m}^{\prime} = {- {\int{r^{\ell}Y_{\quad{\ell\quad m}}^{*}{\nabla{\cdot M}}{\mathbb{d}^{3}x}}}}}\end{matrix} & (9.48)\end{matrix}$From Eq. (1.108), the geometrical factor of the surface current-densityfunction of the orbitsphere about the z-axis is$\left( \frac{2}{3} \right)^{- 1}.$Using the geometrical factor, Eqs. (9.47-9.48), and Eqs. (16.101) and(16.102) of Jackson [36], the multipole coefficient α_(Mag)(l,m) of themagnetic force of Eq. (7.15) is $\begin{matrix}{{\alpha_{Mag}\left( {\ell,m} \right)} = {\frac{\frac{3}{2}}{\left( {{2\ell} + 1} \right)!!}\frac{1}{\ell + 2}\left( \frac{\ell + 1}{\ell} \right)^{1/2}}} & (9.49)\end{matrix}$For singlet states with l≠0, a minimum energy is achieved withconservation of the photon's angular momentum of

when the magnetic moments of the corresponding angular momenta relativeto the electron velocity (and corresponding Lorentzian forces given byEq. (7.5)) superimpose negatively such that the spin component is radial(i_(r)-direction) and the orbital component is central(−i_(r)-direction). The amplitude of the orbital angular momentumL_(rotational orbital), given by Eq. (1.96b) is $\begin{matrix}{L = {{I\quad\omega\quad i_{z}} = {{\hslash\left\lbrack \frac{\ell\left( {\ell + 1} \right)}{\ell^{2} + {2\ell} + 1} \right\rbrack}^{\frac{1}{2}} = \sqrt[\hslash]{\frac{\ell}{\ell + 1}\quad}}}} & (9.50)\end{matrix}$Thus, using Eqs. (7.15), (9.8), and (9.49-9.50), the magnetic forcebetween the two electrons is $\begin{matrix}{F_{Mag} = \begin{matrix}{{- \frac{1}{n}}\frac{\frac{3}{2}}{\left( {{2\ell} + 1} \right)!!}\frac{1}{\ell + 2}\left( \frac{\ell + 1}{\ell} \right)^{1/2}\frac{1}{2}\frac{\hslash^{2}}{m_{e}r^{3}}} \\\left( {\sqrt{s\left( {s + 1} \right)} - \sqrt{\frac{\ell}{\ell + 1}}} \right)\end{matrix}} & (9.51)\end{matrix}$and the force balance equation from Eq. (7.18) which achieves thecondition that the sum of the mechanical momentum and electromagneticmomentum is conserved as given in Sections 6.6, 12.10, and 17.3 ofJackson [37] is $\begin{matrix}{\begin{matrix}{\frac{m_{e}v^{2}}{r_{2}} = {\frac{\hslash^{2}}{m_{e}r_{2}^{3}} = {{\frac{1}{n}\frac{{\mathbb{e}}^{2}}{4{\pi ɛ}_{o}r_{2}^{2}}} - {\frac{1}{n}\frac{\frac{3}{2}}{\left( {{2\ell} + 1} \right)!!}\left( \frac{\ell + 1}{\ell} \right)^{1/2}}}}} \\{\frac{1}{\ell + 2}\frac{1}{2}\frac{\hslash^{2}}{m_{e}r^{3}}\left( {\sqrt{s\left( {s + 1} \right)} - \sqrt{\frac{\ell}{\ell + 1}}} \right)}\end{matrix}{{{{With}\quad s} = \frac{1}{2}},}} & (9.52) \\{{r_{2} = {\left\lbrack {n + {\frac{\frac{3}{4}}{\left( {{2\ell} + 1} \right)!!}\frac{1}{\ell + 2}\left( \frac{\ell + 1}{\ell\quad} \right)^{1/2}\left( {\sqrt{\frac{3}{4}} - \sqrt{\frac{\ell}{\ell + 1}}} \right)}} \right\rbrack\alpha_{He}}}{{n = 2},3,4,\ldots}} & (9.53)\end{matrix}$The excited-state energy is the energy stored in the electric field,E_(ele), given by Eq. (9.12) where r₂ is given by Eq. (9.53). Theenergies of the various singlet excited states of helium with l≠0 appearin TABLE V.

Using r₂ (Eq. (9.53), r₁ can be solved using the equal and oppositemagnetic force of electron 2 on electron 1 and the central Coulombicforce corresponding to the nuclear charge of 2e. Using Eq. (9.52), theforce balance between the centrifugal and electric and magnetic forcesis $\begin{matrix}{\begin{matrix}{\frac{m_{e}v^{2}}{r_{1}} = {\frac{\hslash^{2}}{m_{e}r_{1}^{3}} = {\frac{2{\mathbb{e}}^{2}}{4{\pi ɛ}_{o}r_{1}^{2}} + {\frac{1}{n}\frac{\frac{3}{2}}{\left( {{2\ell} + 1} \right)!!}\left( \frac{\ell + 1}{\ell} \right)^{1/2}}}}} \\{\frac{1}{\ell + 2}\frac{1}{2}\frac{\hslash^{2}}{m_{e}r_{2}^{3}}\left( {\sqrt{s\left( {s + 1} \right)} - \sqrt{\frac{\ell}{\ell + 1}}} \right)}\end{matrix}{{{{With}\quad s} = \frac{1}{2}},}} & (9.54) \\{\begin{matrix}{r_{1}^{3} + {\frac{n\quad 8r_{1}r_{2}^{3}}{3\left( {\sqrt{\frac{3}{4}} - \sqrt{\frac{\ell}{\ell + 1}}} \right)}{\left( {{2\ell} + 1} \right)!!}\left( \frac{\ell}{\ell + 1} \right)^{{1/2}\quad}\left( {\ell + 2} \right)} -} \\{{\frac{n\quad 4r_{2}^{3}}{3\left( {\sqrt{\frac{3}{4}} - \sqrt{\frac{\ell}{\ell + 1}}} \right)}{\left( {{2\ell} + 1} \right)!!}\left( \frac{\ell}{\ell + 1} \right)^{{1/2}\quad}\left( {\ell + 2} \right)} = 0}\end{matrix}{{n = 2},3,4,\ldots}} & (9.55)\end{matrix}$where r₂ is given by Eq. (9.53) and r₁ and r₂ are in units of α_(He). Toobtain the solution of cubic Eq. (9.55), let $\begin{matrix}{g = {{- \frac{n\quad 4r_{2}^{3}}{3\left( {\sqrt{\frac{3}{4}} - \sqrt{\frac{\ell}{\ell + 1}}} \right)}}{\left( {{2\ell} + 1} \right)!!}\left( \frac{\ell}{\ell + 1} \right)^{1/2}\left( {\ell + 2} \right)}} & (9.56)\end{matrix}$Then, Eq. (9.55) becomesr ₁ ³−2gr ₁ +g=0 n=2, 3, 4, . . .   (9.57)Three distinct cases arise depending on the value of l. For l=1 or l=2,g of Eq. (9.56) is negative and A and B of Eqs. (9.20) and (9.21),respectively, are real: $\begin{matrix}{{A = {\sqrt[3]{- \frac{g}{2}}\sqrt[3]{1 + \sqrt{1 - {\frac{32}{27}g}}}}}{and}} & (9.58) \\{B = {{- \sqrt[3]{- \frac{g}{2}}}\sqrt[3]{1 + \sqrt{1 - {\frac{32}{27}g}} - 1}}} & (9.59)\end{matrix}$The only real root is $\begin{matrix}{r_{1} = {r_{11} = {\sqrt[3]{- \frac{g}{2}}\begin{Bmatrix}{\sqrt[3]{1 + \sqrt{1 - {\frac{32}{27}g}}} -} \\\sqrt[3]{\sqrt{1 - {\frac{32}{27}g}} - 1}\end{Bmatrix}}}} & (9.60)\end{matrix}$while r₁₂ and r₁₃ are complex conjugates. When l=3 the magnetic forceterm (2nd term on RHS) of Eq. (9.52) is zero, and the force balancetrivially givesr₁=0.5α_(He)  (9.61)

When l=4, 5, 6 . . . all three roots are real, but, the physical root isr₁₃. In this case, note that n≧5, l≧4; so, the factor g of Eq. (9.56) islarge (>10⁸). Expanding r₁₃ for large values of g gives $\begin{matrix}{r_{1} = {r_{13} = {\frac{1}{2} + \frac{1}{16g} + {O\left( g^{{- 3}/2} \right)}}}} & (9.62)\end{matrix}$ TABLE V Calculated and experimental energies of He Isinglet excited states with l ≠ 0. E_(ele) CQM NIST He I Energy He IEnergy Difference Relative r₁ r₂ Term Levels ^(c) Levels ^(d) CQM − NISTDifference ^(e) n l (α_(He)) ^(a) (α_(He)) ^(b) Symbol (eV) (eV) (eV)(CQM − NIST) 2 1 0.499929 2.01873 1s2p ¹P⁰ −3.36941 −3.36936 −0.00004770.0000141 3 2 0.499999 3.00076 1s3d ¹D −1.51116 −1.51331 0.0021542−0.0014235 3 1 0.499986 3.01873 1s3p ¹P⁰ −1.50216 −1.50036 −0.00179990.0011997 4 2 0.500000 4.00076 1s4d ¹D −0.85008 −0.85105 0.0009711−0.0011411 4 3 0.500000 4.00000 1s4f ¹F⁰ −0.85024 −0.85037 0.0001300−0.0001529 4 1 0.499995 4.01873 1s4p ¹P⁰ −0.84628 −0.84531 −0.00096760.0011446 5 2 0.500000 5.00076 1s5d ¹D −0.54407 −0.54458 0.0005089−0.0009345 5 3 0.500000 5.00000 1s5f ¹F⁰ −0.54415 −0.54423 0.0000764−0.0001404 5 4 0.500000 5.00000 1s5g ¹G −0.54415 −0.54417 0.0000159−0.0000293 5 1 0.499998 5.01873 1s5p ¹P⁰ −0.54212 −0.54158 −0.00054290.0010025 6 2 0.500000 6.00076 1s6d ¹D −0.37784 −0.37813 0.0002933−0.0007757 6 3 0.500000 6.00000 1s6f ¹F⁰ −0.37788 −0.37793 0.0000456−0.0001205 6 4 0.500000 6.00000 1s6g ¹G −0.37788 −0.37789 0.0000053−0.0000140 6 5 0.500000 6.00000 1s6h ¹H⁰ −0.37788 −0.37788 −0.00000450.0000119 6 1 0.499999 6.01873 1s6p ¹P⁰ −0.37671 −0.37638 −0.00032860.0008730 7 2 0.500000 7.00076 1s7d ¹D −0.27760 −0.27779 0.0001907−0.0006864 7 3 0.500000 7.00000 1s7f ¹F⁰ −0.27763 −0.27766 0.0000306−0.0001102 7 4 0.500000 7.00000 1s7g ¹G −0.27763 −0.27763 0.0000004−0.0000016 7 5 0.500000 7.00000 1s7h ¹H⁰ −0.27763 −0.27763 0.0000006−0.0000021 7 6 0.500000 7.00000 1s7i ¹I −0.27763 −0.27762 −0.00000940.0000338 7 1 0.500000 7.01873 1s7p ¹P⁰ −0.27689 −0.27667 −0.00021860.0007900 Avg. 0.0000240 −0.0000220^(a) Radius of the inner electron 1 from Eq. (9.60) for l = 1 or l = 2,Eq. (9.61) for l = 3, and Eq. (9.62) for l = 4, 5, 6 . . . .^(b) Radius of the outer electron 2 from Eq. (9.53).^(e) Classical quantum mechanical (CQM) calculated energy levels givenby the electric energy (Eq. (9.12)).^(d) Experimental NIST levels [34] with the ionization potential definedas zero.^(e) (Theoretical-Experimental)/Experimental.3.D Triplet Excited States with l≠0

For triplet states with l≠0, a minimum energy is achieved withconservation of the photon's angular momentum of

when the magnetic moments of the corresponding angular momentasuperimpose negatively such that the spin component is central and theorbital component is radial. Furthermore, as given for the tripletstates with l=0, the spin component in Eqs. (9.51) and (9.52) isdoubled. Thus, the force balance equation is given by $\begin{matrix}{\begin{matrix}{\frac{m_{e}v^{2}}{r_{2}} = {\frac{\hslash^{2}}{m_{e}r_{2}^{3}} = {{\frac{1}{n}\frac{e^{2}}{4{\pi ɛ}_{o}r_{2}^{2}}} + {\frac{1}{n}\frac{\frac{3}{2}}{\left( {{2\ell} + 1} \right)!!}\left( \frac{\ell + 1}{\ell} \right)^{1/2}}}}} \\{\frac{1}{\ell + 2}\frac{1}{2}\frac{\hslash^{2}}{m_{e}r^{3}}\left( {{2\sqrt{s\left( {s + 1} \right)}} - \sqrt{\frac{\ell}{\ell + 1}}} \right)}\end{matrix}{{{{With}\quad s} = \frac{1}{2}},}} & (9.63) \\{{r_{2} = {\left\lbrack {n - {\frac{\frac{3}{4}}{\left( {{2\ell} + 1} \right)!!}\frac{1}{\ell + 2}\left( \frac{\ell + 1}{\ell\quad} \right)^{1/2}\left( {{2\sqrt{\frac{3}{4}}} - \sqrt{\frac{\ell}{\ell + 1}}} \right)}} \right\rbrack\alpha_{He}}}{{n = 2},3,4,\ldots}} & (9.64)\end{matrix}$The excited-state energy is the energy stored in the electric field,E_(ele), given by Eq. (9.12) where r₂ is given by Eq. (9.64). Theenergies of the various triplet excited states of helium with l≠0 appearin TABLE VI.

Using r₂ (Eq. (9.64), r₁ can be solved using the equal and oppositemagnetic force of electron 2 on electron 1 and the central Coulombicforce corresponding to the nuclear charge of 2e. Using Eq. (9.63), theforce balance between the centrifugal and electric and magnetic forcesis $\begin{matrix}{\begin{matrix}{\frac{m_{e}v^{2}}{r_{1}} = {\frac{\hslash^{2}}{m_{e}r_{1}^{3}} = {\frac{2e^{2}}{4{\pi ɛ}_{o}r_{1}^{2}} - {\frac{1}{n}\frac{\frac{3}{2}}{\left( {{2\ell} + 1} \right)!!}\left( \frac{\ell + 1}{\ell} \right)^{1/2}}}}} \\{\frac{1}{\ell + 2}\frac{1}{2}\frac{\hslash^{2}}{m_{e}r_{2}^{3}}\left( {\sqrt[2]{s\left( {s + 1} \right)} - \sqrt{\frac{\ell}{\ell + 1}}} \right)}\end{matrix}{{{{With}\quad s} = \frac{1}{2}},}} & (9.65) \\{\begin{matrix}{r_{1}^{3} + {\frac{n\quad 8r_{1}r_{2}^{3}}{3\left( {\sqrt{\frac{3}{4}} - \sqrt{\frac{\ell}{\ell + 1}}} \right)}{\left( {{2\ell} + 1} \right)!!}\left( \frac{\ell}{\ell + 1} \right)^{{1/2}\quad}\left( {\ell + 2} \right)} +} \\{{\frac{n\quad 4r_{2}^{3}}{3\left( {\sqrt{\frac{3}{4}} - \sqrt{\frac{\ell}{\ell + 1}}} \right)}{\left( {{2\ell} + 1} \right)!!}\left( \frac{\ell}{\ell + 1} \right)^{{1/2}\quad}\left( {l + 2} \right)} = 0}\end{matrix}{{n = 2},3,4,\ldots}} & (9.66)\end{matrix}$where r₂ is given by Eq. (9.64) and r₁ and r₂ are in units of α_(He). Toobtain the solution of cubic Eq. (9.66), let $\begin{matrix}{g = {\frac{n\quad 4r_{2}^{3}}{3\left( {\sqrt{3} - \sqrt{\frac{\ell}{\ell + 1}}} \right)}{\left( {{2\ell} + 1} \right)!!}\left( \frac{\ell}{\ell + 1} \right)^{1/2}\left( {\ell + 2} \right)}} & (9.67)\end{matrix}$Then, Eq. (9.66) becomesr ₁ ³−2gr ₁ +g=0 n=2, 3, 4, . . .   (9.68)

Using Eqs. (9.16-9.29), the real and physical root is $\begin{matrix}{r_{1} = {r_{13} = {{- \sqrt{\frac{2}{3}g}}\left( {{\cos\frac{\theta}{3}} - {\sqrt{3}\sin\frac{\theta}{3}}} \right)}}} & (9.69)\end{matrix}$ TABLE VI Calculated and experimental energies of He Itriplet excited states with l ≠ 0. E_(ele) CQM NIST He I Energy He IEnergy Difference Relative r₁ r₂ Term Levels ^(c) Levels ^(d) CQM − NISTDifference ^(e) n l (α_(He)) ^(a) (α_(He)) ^(b) Symbol (eV) (eV) (eV)(CQM − NIST) 2 1 0.500571 1.87921 1s2p ³P⁰ ₂ −3.61957 −3.6233 0.0037349−0.0010308 2 1 0.500571 1.87921 1s2p ³P⁰ ₁ −3.61957 −3.62329 0.0037249−0.0010280 2 1 0.500571 1.87921 1s2p ³P⁰ ₀ −3.61957 −3.62317 0.0036049−0.0009949 3 1 0.500105 2.87921 1s3p ³P⁰ ₂ −1.57495 −1.58031 0.0053590−0.0033911 3 1 0.500105 2.87921 1s3p ³P⁰ ₁ −1.57495 −1.58031 0.0053590−0.0033911 3 1 0.500105 2.87921 1s3p ³P⁰ ₀ −1.57495 −1.58027 0.0053190−0.0033659 3 2 0.500011 2.98598 1s3d ³D₃ −1.51863 −1.51373 −0.00490310.0032391 3 2 0.500011 2.98598 1s3d ³D₂ −1.51863 −1.51373 −0.00490310.0032391 3 2 0.500011 2.98598 1s3d ³D₁ −1.51863 −1.51373 −0.00490310.0032391 4 1 0.500032 3.87921 1s4p ³P⁰ ₂ −0.87671 −0.87949 0.0027752−0.0031555 4 1 0.500032 3.87921 1s4p ³P⁰ ₁ −0.87671 −0.87949 0.0027752−0.0031555 4 1 0.500032 3.87921 1s4p ³P⁰ ₀ −0.87671 −0.87948 0.0027652−0.0031442 4 2 0.500003 3.98598 1s4d ³D₃ −0.85323 −0.85129 −0.00193980.0022787 4 2 0.500003 3.98598 1s4d ³D₂ −0.85323 −0.85129 −0.00193980.0022787 4 2 0.500003 3.98598 1s4d ³D₁ −0.85323 −0.85129 −0.00193980.0022787 4 3 0.500000 3.99857 1s4f³F⁰ ₃ −0.85054 −0.85038 −0.00016380.0001926 4 3 0.500000 3.99857 1s4f³F⁰ ₄ −0.85054 −0.85038 −0.00016380.0001926 4 3 0.500000 3.99857 1s4f³F⁰ ₂ −0.85054 −0.85038 −0.00016380.0001926 5 1 0.500013 4.87921 1s5p ³P⁰ ₂ −0.55762 −0.55916 0.0015352−0.0027456 5 1 0.500013 4.87921 1s5p ³P⁰ ₁ −0.55762 −0.55916 0.0015352−0.0027456 5 1 0.500013 4.87921 1s5p ³P⁰ ₀ −0.55762 −0.55915 0.0015252−0.0027277 5 2 0.500001 4.98598 1s5d ³D₃ −0.54568 −0.54472 −0.00096330.0017685 5 2 0.500001 4.98598 1s5d ³D₂ −0.54568 −0.54472 −0.00096330.0017685 5 2 0.500001 4.98598 1s5d ³D₁ −0.54568 −0.54472 −0.00096330.0017685 5 3 0.500000 4.99857 1s5f³F⁰ ₃ −0.54431 −0.54423 −0.00007910.0001454 5 3 0.500000 4.99857 1s5f³F⁰ ₄ −0.54431 −0.54423 −0.00007910.0001454 5 3 0.500000 4.99857 1s5f³F⁰ ₂ −0.54431 −0.54423 −0.00007910.0001454 5 4 0.500000 4.99988 1s5g ³G₄ −0.54417 −0.54417 0.0000029−0.0000054 5 4 0.500000 4.99988 1s5g ³G₅ −0.54417 −0.54417 0.0000029−0.0000054 5 4 0.500000 4.99988 1s5g ³G₃ −0.54417 −0.54417 0.0000029−0.0000054 6 1 0.500006 5.87921 1s6p ³P⁰ ₂ −0.38565 −0.38657 0.0009218−0.0023845 6 1 0.500006 5.87921 1s6p ³P⁰ ₁ −0.38565 −0.38657 0.0009218−0.0023845 6 1 0.500006 5.87921 1s6p ³P⁰ ₀ −0.38565 −0.38657 0.0009218−0.0023845 6 2 0.500001 5.98598 1s6d ³D₃ −0.37877 −0.37822 −0.00054930.0014523 6 2 0.500001 5.98598 1s6d ³D₂ −0.37877 −0.37822 −0.00054930.0014523 6 2 0.500001 5.98598 1s6d ³D₁ −0.37877 −0.37822 −0.00054930.0014523 6 3 0.500000 5.99857 1s6f³F⁰ ₃ −0.37797 −0.37793 −0.00004440.0001176 6 3 0.500000 5.99857 1s6f³F⁰ ₄ −0.37797 −0.37793 −0.00004440.0001176 6 3 0.500000 5.99857 1s6f³F⁰ ₂ −0.37797 −0.37793 −0.00004440.0001176 6 4 0.500000 5.99988 1s6g ³G₄ −0.37789 −0.37789 −0.00000230.0000060 6 4 0.500000 5.99988 1s6g ³G₅ −0.37789 −0.37789 −0.00000230.0000060 6 4 0.500000 5.99988 1s6g ³G₃ −0.37789 −0.37789 −0.00000230.0000060 6 5 0.500000 5.99999 1s6h ³H⁰ ₄ −0.37789 −0.37788 −0.00000500.0000133 6 5 0.500000 5.99999 1s6h ³H⁰ ₅ −0.37789 −0.37788 −0.00000500.0000133 6 5 0.500000 5.99999 1s6h ³H⁰ ₆ −0.37789 −0.37788 −0.00000500.0000133 7 1 0.500003 6.87921 1s7p ³P⁰ ₂ −0.28250 −0.28309 0.0005858−0.0020692 7 1 0.500003 6.87921 1s7p ³P⁰ ₁ −0.28250 −0.28309 0.0005858−0.0020692 7 1 0.500003 6.87921 1s7p ³P⁰ ₀ −0.28250 −0.28309 0.0005858−0.0020692 7 2 0.500000 6.98598 1s7d ³D₃ −0.27819 −0.27784 −0.00034640.0012468 7 2 0.500000 6.98598 1s7d ³D₂ −0.27819 −0.27784 −0.00034640.0012468 7 2 0.500000 6.98598 1s7d ³D₁ −0.27819 −0.27784 −0.00034640.0012468 7 3 0.500000 6.99857 1s7f³F⁰ ₃ −0.27769 −0.27766 −0.00002610.0000939 7 3 0.500000 6.99857 1s7f³F⁰ ₄ −0.27769 −0.27766 −0.00002610.0000939 7 3 0.500000 6.99857 1s7f³F⁰ ₂ −0.27769 −0.27766 −0.00002610.0000939 7 4 0.500000 6.99988 1s7g ³G₄ −0.27763 −0.27763 −0.00000430.0000155 7 4 0.500000 6.99988 1s7g ³G₅ −0.27763 −0.27763 −0.00000430.0000155 7 4 0.500000 6.99988 1s7g ³G₃ −0.27763 −0.27763 −0.00000430.0000155 7 5 0.500000 6.99999 1s7h ³H⁰ ₅ −0.27763 −0.27763 0.0000002−0.0000009 7 5 0.500000 6.99999 1s7h ³H⁰ ₆ −0.27763 −0.27763 0.0000002−0.0000009 7 5 0.500000 6.99999 1s7h ³H⁰ ₄ −0.27763 −0.27763 0.0000002−0.0000009 7 6 0.500000 7.00000 1s7 i³I₅ −0.27763 −0.27762 −0.00000940.0000339 7 6 0.500000 7.00000 1s7i ³I6 −0.27763 −0.27762 −0.00000940.0000339 7 6 0.500000 7.00000 1s7i ³I₇ −0.27763 −0.27762 −0.00000940.0000339 Avg. 0.0002768 −0.0001975^(a) Radius of the inner electron 1 from Eq. (9.69).^(b) Radius of the outer electron 2 from Eq. (9.64).^(e) Classical quantum mechanical (CQM) calculated energy levels givenby the electric energy (Eq. (9.12)).^(d) Experimental NIST levels [34] with the ionization potential definedas zero.^(e)(Theoretical-Experimental)/Experimental.3.E All Excited He I States

The combined energies of the various states of helium appear in TABLEVII. A plot of the predicted and experimental energies of levelsassigned by NIST [34] appears in FIG. 6. For over 100 states, ther-squared value is 0.999994, and the typical average relative differenceis about 5 significant figures which is within the error of theexperimental data. The agreement is remarkable.

The hydrino states given in the Hydrino Theory—BlackLight Processsection of Ref. [1] are strongly supported by the calculation of thehelium excited states as well as the hydrogen excited states given inthe Excited States of the One-Electron Atom (Quantization) section ofRef. [1] since the electron-photon model is the same in both theexcited-states and in the lower-energy states of hydrogen except thatthe photon provides a central field of magnitude n in the hydrino caseand 1/n in the excited-state case. TABLE VII Calculated and experimentalenergies of states of helium. E_(ele) CQM NIST He I Energy He I EnergyDifference Relative r₁ r₂ Term Levels ^(c) Levels ^(d) CQM − NISTDifference ^(e) n l (α_(He)) ^(a) (α_(He)) ^(b) Symbol (eV) (eV) (eV)(CQM − NIST) 1 0 0.56699 0.566987 1s² ¹S −24.58750 −24.58741 −0.0000920.0000038 2 0 0.506514 1.42265 1s2s ³S −4.78116 −4.76777 −0.01339290.0028090 2 0 0.501820 1.71132 1s2s ¹S −3.97465 −3.97161 −0.00304160.0007658 2 1 0.500571 1.87921 1s2p ³P⁰ ₂ −3.61957 −3.6233 0.0037349−0.0010308 2 1 0.500571 1.87921 1s2p ³P⁰ ₁ −3.61957 −3.62329 0.0037249−0.0010280 2 1 0.500571 1.87921 1s2p ³P⁰ ₀ −3.61957 −3.62317 0.0036049−0.0009949 2 1 0.499929 2.01873 1s2p ¹P⁰ −3.36941 −3.36936 −0.00004770.0000141 3 0 0.500850 2.42265 1s3s ³S −1.87176 −1.86892 −0.00283770.0015184 3 0 0.500302 2.71132 1s3s ¹S −1.67247 −1.66707 −0.00540140.0032401 3 1 0.500105 2.87921 1s3p ³P⁰ ₂ −1.57495 −1.58031 0.0053590−0.0033911 3 1 0.500105 2.87921 1s3p ³P⁰ ₁ −1.57495 −1.58031 0.0053590−0.0033911 3 1 0.500105 2.87921 1s3p ³P⁰ ₀ −1.57495 −1.58027 0.0053190−0.0033659 3 2 0.500011 2.98598 1s3d ³D₃ −1.51863 −1.51373 −0.00490310.0032391 3 2 0.500011 2.98598 1s3d ³D₂ −1.51863 −1.51373 −0.00490310.0032391 3 2 0.500011 2.98598 1s3d ³D₁ −1.51863 −1.51373 −0.00490310.0032391 3 2 0.499999 3.00076 1s3d ¹D −1.51116 −1.51331 0.0021542−0.0014235 3 1 0.499986 3.01873 1s3p ¹P⁰ −1.50216 −1.50036 −0.00179990.0011997 4 0 0.500225 3.42265 1s4s ³S −0.99366 −0.99342 −0.00024290.0002445 4 0 0.500088 3.71132 1s4s ¹S −0.91637 −0.91381 −0.00256360.0028054 4 1 0.500032 3.87921 1s4p ³P⁰ ₂ −0.87671 −0.87949 0.0027752−0.0031555 4 1 0.500032 3.87921 1s4p ³P⁰ ₁ −0.87671 −0.87949 0.0027752−0.0031555 4 1 0.500032 3.87921 1s4p ³P⁰ ₀ −0.87671 −0.87948 0.0027652−0.0031442 4 2 0.500003 3.98598 1s4d ³D₃ −0.85323 −0.85129 −0.00193980.0022787 4 2 0.500003 3.98598 1s4d ³D₂ −0.85323 −0.85129 −0.00193980.0022787 4 2 0.500003 3.98598 1s4d ³D₁ −0.85323 −0.85129 −0.00193980.0022787 4 2 0.500000 4.00076 1s4d ¹D −0.85008 −0.85105 0.0009711−0.0011411 4 3 0.500000 3.99857 1s4f³F⁰ ₃ −0.85054 −0.85038 −0.00016380.0001926 4 3 0.500000 3.99857 1s4f³F⁰ ₄ −0.85054 −0.85038 −0.00016380.0001926 4 3 0.500000 3.99857 1s4f³F⁰ ₂ −0.85054 −0.85038 −0.00016380.0001926 4 3 0.500000 4.00000 1s4f¹F⁰ −0.85024 −0.85037 0.0001300−0.0001529 4 1 0.499995 4.01873 1s4p ¹P⁰ −0.84628 −0.84531 −0.00096760.0011446 5 0 0.500083 4.42265 1s5s ³S −0.61519 −0.61541 0.0002204−0.0003582 5 0 0.500035 4.71132 1s5s ¹S −0.57750 −0.57617 −0.00132530.0023002 5 1 0.500013 4.87921 1s5p ³P⁰ ₂ −0.55762 −0.55916 0.0015352−0.0027456 5 1 0.500013 4.87921 1s5p ³P⁰ ₁ −0.55762 −0.55916 0.0015352−0.0027456 5 1 0.500013 4.87921 1s5p ³P⁰ ₀ −0.55762 −0.55915 0.0015252−0.0027277 5 2 0.500001 4.98598 1s5d ³D₃ −0.54568 −0.54472 −0.00096330.0017685 5 2 0.500001 4.98598 1s5d ³D₂ −0.54568 −0.54472 −0.00096330.0017685 5 2 0.500001 4.98598 1s5d ³D₁ −0.54568 −0.54472 −0.00096330.0017685 5 2 0.500000 5.00076 1s5d ¹D −0.54407 −0.54458 0.0005089−0.0009345 5 3 0.500000 4.99857 1s5f³F⁰ ₃ −0.54431 −0.54423 −0.00007910.0001454 5 3 0.500000 4.99857 1s5f³F⁰ ₄ −0.54431 −0.54423 −0.00007910.0001454 5 3 0.500000 4.99857 1s5f³F⁰ ₂ −0.54431 −0.54423 −0.00007910.0001454 5 3 0.500000 5.00000 1s5f¹F⁰ −0.54415 −0.54423 0.0000764−0.0001404 5 4 0.500000 4.99988 1s5g ³G₄ −0.54417 −0.54417 0.0000029−0.0000054 5 4 0.500000 4.99988 1s5g ³G₅ −0.54417 −0.54417 0.0000029−0.0000054 5 4 0.500000 4.99988 1s5g ³G₃ −0.54417 −0.54417 0.0000029−0.0000054 5 4 0.500000 5.00000 1s5g ¹G −0.54415 −0.54417 0.0000159−0.0000293 5 1 0.499998 5.01873 1s5p ¹P⁰ −0.54212 −0.54158 −0.00054290.0010025 6 0 0.500038 5.42265 1s6s ³S −0.41812 −0.41838 0.0002621−0.0006266 6 0 0.500016 5.71132 1s6s ¹S −0.39698 −0.39622 −0.00076440.0019291 6 1 0.500006 5.87921 1s6p ³P⁰ ₂ −0.38565 −0.38657 0.0009218−0.0023845 6 1 0.500006 5.87921 1s6p ³P⁰ ₁ −0.38565 −0.38657 0.0009218−0.0023845 6 1 0.500006 5.87921 1s6p ³P⁰ ₀ −0.38565 −0.38657 0.0009218−0.0023845 6 2 0.500001 5.98598 1s6d ³D₃ −0.37877 −0.37822 −0.00054930.0014523 6 2 0.500001 5.98598 1s6d ³D₂ −0.37877 −0.37822 −0.00054930.0014523 6 2 0.500001 5.98598 1s6d ³D₁ −0.37877 −0.37822 −0.00054930.0014523 6 2 0.500000 6.00076 1s6d ¹D −0.37784 −0.37813 0.0002933−0.0007757 6 3 0.500000 5.99857 1s6f³F⁰ ₃ −0.37797 −0.37793 −0.00004440.0001176 6 3 0.500000 5.99857 1s6f³F⁰ ₄ −0.37797 −0.37793 −0.00004440.0001176 6 3 0.500000 5.99857 1s6f³F⁰ ₂ −0.37797 −0.37793 −0.00004440.0001176 6 3 0.500000 6.00000 1s6f¹F⁰ −0.37788 −0.37793 0.0000456−0.0001205 6 4 0.500000 5.99988 1s6g ³G₄ −0.37789 −0.37789 −0.00000230.0000060 6 4 0.500000 5.99988 1s6g ³G₅ −0.37789 −0.37789 −0.00000230.0000060 6 4 0.500000 5.99988 1s6g ³G₃ −0.37789 −0.37789 −0.00000230.0000060 6 4 0.500000 6.00000 1s6g ¹G −0.37788 −0.37789 0.0000053−0.0000140 6 5 0.500000 5.99999 1s6h ³H⁰ ₄ −0.37789 −0.37788 −0.00000500.0000133 6 5 0.500000 5.99999 1s6h ³H⁰ ₅ −0.37789 −0.37788 −0.00000500.0000133 6 5 0.500000 5.99999 1s6h ³H⁰ ₆ −0.37789 −0.37788 −0.00000500.0000133 6 5 0.500000 6.00000 1s6h ¹H⁰ −0.37788 −0.37788 −0.00000450.0000119 6 1 0.499999 6.01873 1s6p ¹P⁰ −0.37671 −0.37638 −0.00032860.0008730 7 0 0.500019 6.42265 1s7s ³S −0.30259 −0.30282 0.0002337−0.0007718 7 0 0.500009 6.71132 1s7s ¹S −0.28957 −0.2891 −0.00047110.0016295 7 1 0.500003 6.87921 1s7p ³P⁰ ₂ −0.28250 −0.28309 0.0005858−0.0020692 7 1 0.500003 6.87921 1s7p ³P⁰ ₁ −0.28250 −0.28309 0.0005858−0.0020692 7 1 0.500003 6.87921 1s7p ³P⁰ ₀ −0.28250 −0.28309 0.0005858−0.0020692 7 2 0.500000 6.98598 1s7d ³D₃ −0.27819 −0.27784 −0.00034640.0012468 7 2 0.500000 6.98598 1s7d ³D₂ −0.27819 −0.27784 −0.00034640.0012468 7 2 0.500000 6.98598 1s7d ³D₁ −0.27819 −0.27784 −0.00034640.0012468 7 2 0.500000 7.00076 1s7d ¹D −0.27760 −0.27779 0.0001907−0.0006864 7 3 0.500000 6.99857 1s7f³F⁰ ₃ −0.27769 −0.27766 −0.00002610.0000939 7 3 0.500000 6.99857 1s7f³F⁰ ₄ −0.27769 −0.27766 −0.00002610.0000939 7 3 0.500000 6.99857 1s7f³F⁰ ₂ −0.27769 −0.27766 −0.00002610.0000939 7 3 0.500000 7.00000 1s7f ¹F⁰ −0.27763 −0.27766 0.0000306−0.0001102 7 4 0.500000 6.99988 1s7g ³G₄ −0.27763 −0.27763 −0.00000430.0000155 7 4 0.500000 6.99988 1s7g ³G₅ −0.27763 −0.27763 −0.00000430.0000155 7 4 0.500000 6.99988 1s7g ³G₃ −0.27763 −0.27763 −0.00000430.0000155 7 4 0.500000 7.00000 1s7g ¹G −0.27763 −0.27763 0.0000004−0.0000016 7 5 0.500000 6.99999 1s7h ³H⁰ ₅ −0.27763 −0.27763 0.0000002−0.0000009 7 5 0.500000 6.99999 1s7h ³H⁰ ₆ −0.27763 −0.27763 0.0000002−0.0000009 7 5 0.500000 6.99999 1s7h ³H⁰ ₄ −0.27763 −0.27763 0.0000002−0.0000009 7 5 0.500000 7.00000 1s7h ¹H⁰ −0.27763 −0.27763 0.0000006−0.0000021 7 6 0.500000 7.00000 1s7 i³I₅ −0.27763 −0.27762 −0.00000940.0000339 7 6 0.500000 7.00000 1s7i ³I6 −0.27763 −0.27762 −0.00000940.0000339 7 6 0.500000 6.78349 1s7i ³I₇ −0.27763 −0.27762 −0.00000940.0000339 7 6 0.500000 7.00000 1s7i ¹I −0.27763 −0.27762 −0.00000940.0000338 7 1 0.500000 7.01873 1s7p ¹P⁰ −0.27689 −0.27667 −0.00021860.0007900 8 0 0.500011 7.42265 1s8s ³S −0.22909 −0.22928 0.0001866−0.0008139 8 0 0.500005 7.71132 1s8s ¹S −0.22052 −0.2202 −0.00031720.0014407 9 0 0.500007 8.42265 1s9s ³S −0.17946 −0.17961 0.0001489−0.0008291 9 0 0.500003 8.71132 1s9s ¹S −0.17351 −0.1733 −0.00021410.0012355 10 0 0.500004 9.42265 1s10s ³S −0.14437 −0.1445 0.0001262−0.0008732 10 0 0.500002 9.71132 1s10s ¹S −0.14008 −0.13992 −0.00016220.0011594 11 0 0.500003 10.42265 1s11s ³S −0.11866 −0.11876 0.0001037−0.0008734 11 0 0.500001 10.71132 1s11s ¹S −0.11546 −0.11534 −0.00011840.0010268 Avg. −0.000112 0.0000386^(a) Radius of the inner electron 1 of singlet excited states with l = 0from Eq. (9.29); triplet excited states with l = 0 from Eq. (9.37);singlet excited states with l ≠ 0 from Eq. (9.60) for l = 1 or l = 2 andEq. (9.61) for l = 3, and Eq. (9.62) for l = 4, 5, 6 . . . ; tripletexcited states with l ≠ 0 from Eq. (9.69), and 1s² ¹S from Eq. (7.19).^(b) Radius of the outer electron 2 of singlet excited states with l = 0from Eq. (9.11); triplet excited states with l = 0 from Eq. (9.32);singlet excited states with l ≠ 0 from Eq. (9.53); triplet excitedstates with l ≠ 0 from Eq. (9.64), and 1s² ¹S from Eq. (7.19).^(e) Classical quantum mechanical (CQM) calculated excited-state energylevels given by the electric energy (Eq. (9.12)) and the energy level of1s² ¹S is given by Eqs. (7.28-7.30).^(d) Experimental NIST levels [34] with the ionization potential definedas zero.^(e) (Theoretical-Experimental)/Experimental.3.F Spin-Orbital Coupling of Excited States with l≠0

Due to 1.) the invariance of each of $\frac{e}{m_{e}}$of the electron, the electron angular momentum of

, and μ_(B) from the spin angular and orbital angular momentum, 2.) thecondition that flux must be linked by the electron orbitsphere in unitsof the magnetic flux quantum, and 3.) the maximum projection of the spinangular momentum of the electron onto an axis is${\sqrt{\frac{3}{4}}\hslash},$the magnetic energy term of the electron g-factor gives the spin-orbitalcoupling energy E_(s/o (Eq. ()2.102)): $\begin{matrix}{E_{s/o} = {{2\frac{\alpha}{2\pi}\left( \frac{e\quad\hslash}{2m_{e}} \right)\frac{\mu_{0}e\quad\hslash}{2\left( {2\pi\quad m_{e}} \right)\left( \frac{r}{2\pi} \right)^{3}}\sqrt{\frac{3}{4}}} = {\frac{{\alpha\pi\mu}_{0}e^{2}\hslash^{2}}{m_{e}^{2}r^{2}}\sqrt{\frac{3}{4}}}}} & (9.70)\end{matrix}$For the n=2 state of hydrogen, the radius is r=2α₀, and the predictedenergy difference between the ²P_(3/2) and ²P_(1/2) levels of thehydrogen atom due to spin-orbital interaction is $\begin{matrix}{E_{s/o} = {\frac{{\alpha\pi\mu}_{0}{\mathbb{e}}^{2}\hslash^{2}}{8m_{e}^{2}a_{0}^{3}}\sqrt{\frac{3}{4}}}} & (9.71)\end{matrix}$As in the case of the ²P_(1/2)→²S_(1/2) transition, the photon-momentumtransfer for the ²P_(3/2)→²P_(1/2) transition gives rise to a smallfrequency shift derived after that of the Lamb shift with Δm_(t)=−1included. The energy, E_(FS), for the ²P_(3/2)→²P_(1/2) transitioncalled the fine structure splitting is given by Eq. (2.113):$\begin{matrix}\begin{matrix}{E_{FS} = {{\frac{{\alpha^{5}\left( {2\pi} \right)}^{2}}{8}m_{e}c^{2}\sqrt{\frac{3}{4}}} + \left( {13.5983\quad{\mathbb{e}}\quad{V\left( {1 - \frac{1}{2^{2}}} \right)}} \right)^{2}}} \\{\left\lbrack {\frac{\left( {\frac{3}{4\pi}\left( {1 - \sqrt{\frac{3}{4}}} \right)} \right)^{2}}{2h\quad\mu_{e}c^{2}} + \frac{\left( {1 + \left( {1 - \sqrt{\frac{3}{4}}} \right)} \right)^{2}}{{2h\quad m_{H}c^{2}}\quad}} \right\rbrack} \\{= {{4.5190\quad \times \quad 10^{- 5}\quad{\mathbb{e}}\quad V} + {1.75407\quad \times \quad 10^{- 7}\quad{\mathbb{e}}\quad V}}} \\{= {4.53659\quad \times \quad 10^{- 5}\quad{\mathbb{e}}\quad V}}\end{matrix} & (9.72)\end{matrix}$

where the first term corresponds to E_(s/o) given by Eq. (9.71)expressed in terms of the mass energy of the electron (Eq. (2.106)) andthe second and third terms correspond to the electron recoil and atomrecoil, respectively. The energy of 4.53659×10⁻⁵ eV corresponds to afrequency of 10,969.4 MHz or a wavelength of 2.73298 cm. Theexperimental value of the ²P_(3/2)→²P_(1/2) transition frequency is10,969.1 MHz. The large natural widths of the hydrogen 2p levels limitsthe experimental accuracy; yet, given this limitation, the agreementbetween the theoretical and experimental fine structure is excellent.Using r₂ given by Eq. (9.53), the spin-orbital energies were calculatedfor l=1 using Eq. (9.70) to compare to the effect of different l quantumnumbers. There is agreement between the magnitude of the predictedresults given in TABLE VIII and the experimental dependence on the lquantum number as given in TABLE VII. TABLE VIII Calculated spin-orbitalenergies of He I singlet excited states with l = 1 as a function of theradius of the outer electron. E_(slo) r₂ spin-orbital coupling^(b) n(α_(He))^(a) Term Symbol (eV) 2 2.01873 1s2p ¹P⁰ 0.0000439 3 3.018731s3p ¹P⁰ 0.0000131 4 4.01873 1s4p ¹P⁰ 0.0000056 5 5.01873 1s5p ¹P⁰0.0000029 6 6.01873 1s6p ¹P⁰ 0.0000017 7 7.01873 1s7p ¹P⁰ 0.0000010^(a)Radius of the outer electron 2 from Eq. (9.53).^(b)The spin-orbital coupling energy of electron 2 from Eq. (9.70) usingr₂ from Eq. (9.53).4. Systems

Embodiments of the system for performing computing and rendering of thenature of excited-state atomic and atomic-ionic electrons using thephysical solutions may comprise a general purpose computer. Such ageneral purpose computer may have any number of basic configurations.For example, such a general purpose computer may comprise a centralprocessing unit (CPU), one or more specialized processors, systemmemory, a mass storage device such as a magnetic disk, an optical disk,or other storage device, an input means such as a keyboard or mouse, adisplay device, and a printer or other output device. A systemimplementing the present invention can also comprise a special purposecomputer or other hardware system and all should be included within itsscope.

The display can be static or dynamic such that spin and angular motionwith corresponding momenta can be displayed in an embodiment. Thedisplayed information is useful to anticipate reactivity, physicalproperties, and optical absorption and emission. The insight into thenature of atomic and atomic-ionic excited-state electrons can permit thesolution and display of those of other atoms and atomic ions and provideutility to anticipate their reactivity and physical properties as wellas facilitate the development of light sources and materials thatrespond to light.

Embodiments within the scope of the present invention also includecomputer program products comprising computer readable medium havingembodied therein program code means. Such computer readable media can beany available media which can be accessed by a general purpose orspecial purpose computer. By way of example, and not limitation, suchcomputer readable media can comprise RAM, ROM, EPROM, CD ROM, DVD orother optical disk storage, magnetic disk storage or other magneticstorage devices, or any other medium which can embody the desiredprogram code means and which can be accessed by a general purpose orspecial purpose computer. Combinations of the above should also beincluded within the scope of computer readable media. Program code meanscomprises, for example, executable instructions and data which cause ageneral purpose computer or special purpose computer to perform acertain function of a group of functions.

A specific example of the rendering of the electron of atomic hydrogenusing Mathematica and computed on a PC is shown in FIG. 1. The algorithmused was

To generate a spherical shell:

SphericalPlot3D[1,{q,0,p},{f,0,2p},Boxed®False,Axes®False]. Therendering can be viewed from different perspectives. A specific exampleof the rendering of atomic hydrogen using Mathematica and computed on aPC is shown in FIG. 1. The algorithm used was

To generate the picture of the electron and proton:

Electron=SphericalPlot3D[1,{q,0,p},{f,0,2p-p/2},Boxed®False,Axes®False];Proton=Show[Graphics3D[{Blue,PointSize[0.03],Point[{0,0,0}]}],Boxed®False];Show[Electron,Proton];

Specific examples of the rendering of thespherical-and-time-harmonic-electron-charge-density functions ofnon-excited and excited-state electrons using Mathematica and computedon a PC are shown in FIG. 3. The algorithm used was

To generate L1MO:

L1MOcolors[theta_,phi_,det_]=Which[det<0.1333,RGBColor[1.000,0.070,0.079],det<0.2666,RGBColor[1.000,0.369,0.067],det<0.4,RGBColor[1.000,0.681,0.049],det<0.5333,RGBColor[0.984,1.000,0.051],det<0.6666,RGBColor[0.673,1.000,0.058],det<0.8,RGBColor[0.364,1.000,0.055],det<0.9333,RGBColor[0.071,1.000,0.060],det<1.066,RGBColor[0.085,1.000,0.388],det<1.2,RGBColor[0.070,1.000,0.678],det<1.333,RGBColor[0.070,1.000,1.000],det<1.466,RGBColor[0.067,0.698,1.000],det<1.6,RGBColor[0.075,0.401, 1.000],det<1.733,RGBColor[0.067,0.082, 1.000],det<1.866,RGBColor[0.326,0.056,1.000],det£2,RGBColor[0.674,0.079,1.000]];

L1MO=ParametricPlot3D[{Sin [theta] Cos [phi],Sin [theta] SUN [phi],Cos[theta],L1MOcolors[theta,phi,1+Cos [theta]]},{theta,0,Pi},{phi,0,2Pi},Boxed®False,Axes®False,Lighting®False,PlotPoints®{20,20},ViewPoint®{−0.273,−2.030,3.494}];

To generate L1MX:

L1MXcolors[theta_, phi_, det_]=Which [det<0.1333, RGBColor[1.000, 0.070,0.079],det<0.2666, RGBColor[1.000, 0.369, 0.067],det<0.4,RGBColor[1.000, 0.681, 0.049],det<0.5333, RGBColor[0.984, 1.000, 0.051],det<0.6666, RGBColor[0.673, 1.000, 0.058], det<0.8, RGBColor[0.364,1.000, 0.055],det<0.9333, RGBColor[0.071, 1.000, 0.060], det<1.066,RGBColor[0.085, 1.000, 0.388],det<1.2, RGBColor[0.070, 1.000, 0.678],det<1.333, RGBColor[0.070, 1.000, 1.000],det<1.466, RGBColor[0.067,0.698, 1.000], det<1.6, RGBColor[0.075, 0.401, 1.000],det<1.733,RGBColor[0.067, 0.082, 1.000], det<1.866, RGBColor[0.326, 0.056,1.000],det<=2, RGBColor[0.674, 0.079, 1.000]];

L1MX=ParametricPlot3D[{Sin [theta] Cos [phi],Sin [theta] SUN [phi],Cos[theta],L1MXcolors[theta,phi, 1+Sin [theta] Cos [phi]]}, theta,0,Pi},{phi,0,2Pi},Boxed®False,Axes®False,Lighting®False,PlotPoints®{20,20},ViewPoint®{−0.273,−2.030,3.494}];

To generate LIMY:

L1MYcolors[theta_,phi_,det_]=Which[det<0.1333,RGBColor[1.000,0.070,0.079],det<0.2666,RGBColor[1.000,0.369,0.067],det<0.4,RGBColor[1.000,0.681,0.049],det<0.5333,RGBColor[0.984,1.000,0.051],det<0.6666,RGBColor[0.673,1.000,0.058],det<0.8,RGBColor[0.364,1.000,0.055],det<0.9333,RGBColor[0.071,1.000,0.060],det<1.066,RGBColor[0.085,1.000,0.388],det<1.2,RGBColor[0.070,1.000,0.678],det<1.333,RGBColor[0.070,1.000,1.000],det<1.466,RGBColor[0.067,0.698,1.0001,det<1.6,RGBColor[0.075,0.401,1.000],det<1.733,RGBColor[0.067,0.082, 1.000],det<1.866,RGBColor[0.326,0.056,1.000],det£2,RGBColor[0.674,0.079, 1.000]);

L1MY=ParametricPlot3D[{Sin [theta] Cos [phi],Sin [theta] SUN [phi],Cos[theta],L1MYcolors[theta,phi,1+Sin [theta] SUN [phi]]}, {theta,0,Pi},{phi,0,2Pi},Boxed®False,Axes®False,Lighting®False,PlotPoints®{20,20}];

To generate L2MO:

L2MOcolors[theta_, phi_, det_]=Which [det<0.2, RGBColor[1.000, 0.070,0.079],det<0.4, RGBColor[1.000, 0.369, 0.067],det<0.6, RGBColor[1.000,0.681, 0.049],det<0.8, RGBColor[0.984, 1.000, 0.051],det<1,RGBColor[0.673, 1.000, 0.058],det<1.2, RGBColor[0.364, 1.000,0.055],det<1.4, RGBColor[0.071, 1.000, 0.060],det<1.6, RGBColor[0.085,1.000, 0.388],det<1.8, RGBColor[0.070, 1.000, 0.678],det<2,RGBColor[0.070, 1.000, 1.000],det<2.2, RGBColor[0.067, 0.698,1.000],det<2.4, RGBColor[0.075, 0.401, 1.000],det<2.6, RGBColor[0.067,0.082, 1.000],det<2.8, RGBColor[0.326, 0.056, 1.000],det<=3,RGBColor[0.674, 0.079, 1.000]];

L2MO=ParametricPlot3D[{Sin [theta] Cos [phi], Sin [theta] Sin [phi], Cos[theta],

-   -   L2MOcolors[theta, phi, 3 Cos [theta] Cos [theta]]},    -   {theta, 0, Pi}, {phi, 0, 2Pi},    -   Boxed->False, Axes->False, Lighting->False,    -   PlotPoints->{20, 20},    -   ViewPoint->{−0.273, −2.030, 3.494}];        To generate L2MF:        L2MFcolors[theta_,phi_,det_=Which        [det<0.1333,RGBColor[1.000,0.070,0.079],det<0.2666,RGBColor[1.000,0.369,0.067],det<0.4,RGBColor[1.000,0.681,0.049],det<0.5333,RGBColor[0.984,1.000,0.051],det<0.6666,RGBColor[0.673,1.000,0.058],det<0.8,RGBColor[0.364,1.000,0.055],det<0.9333,RGBColor[0.071,1.000,0.060],det<1.066,RGBColor[0.085,1.000,0.388],det<1:2,RGBColor[0.070,1.000,0.678],det<1.333,RGBColor[0.070,        1.000,1.000],det<1.466,RGBColor[0.067,0.698,        1.000],det<]0.6,RGBColor[0.075,0.401,1.000],        det<1.733,RGBColor[0.067,0.082,        1.000],det<1.866,RGBColor[0.326,0.056,        1.000],det£2,RGBColor[0.674,0.079, 1.000]];        L2MF=ParametricPlot3D[{Sin [theta] Cos [phi],Sin [theta] SUN        [phi],Cos [theta],L2MFcolors[theta,phi,1+0.72618 Sin [theta] Cos        [phi] 5 Cos [theta] Cos [theta]-0.72618 Sin [theta] Cos [phi]]},        {theta,0,Pi},{phi,0,2Pi},Boxed®False,Axes®False,Lighting®False,PlotPoints®{20,20},ViewPoint®{−0.273,−2.030,2.494}];        To generate L2MX2Y2:        L2MX2Y2colors[theta_,phi_,det_]=Which        [det<0.1333,RGBColor[1.000,0.070,0.079],det<0.2666,RGBColor[1.000,0.369,0.067],det<0.4,RGBColor[1.000,0.681,0.049],det<0.5333,RGBColor[0.984,1.000,0.051],det<0.6666,RGBColor[0.673,1.000,0.058],det<0.8,RGBColor[0.364,1.000,0.055],det<0.9333,RGBColor[0.071,1.000,0.060],det<1.066,RGBColor[0.085,1.000,0.388],det<1.2,RGBColor[0.070,1.000,0.678],det<1.333,RGBColor[0.070,1.000,1.000],det<1.466,RGBColor[0.067,0.698,        1.000],det<1.6,RGBColor[0.075,0.401,1.000],det<1.733,RGBColor[0.067,0.082,        1.000],det<1.866,RGBColor[0.326,0.056,        1.000],det2,RGBColor[0.674,0.079,1.000]];        L2MX2Y2=ParametricPlot3D[Sin [theta] Cos [phi],Sin [theta] SUN        [phi],Cos [theta],L2MX2Y2colors[theta,phi,1+Sin [theta] Sin        [theta] Cos [2 phi]]},        {theta,0,Pi},{phi,0,2Pi},Boxed®False,Axes®False,Lighting®False,PlotPoints®{20,20},ViewPoint®{−0.273,−2.030,3.494}];        To generate L2MXY:        L2MXYcolors[theta_,phi_,det_]=Which        [det<0.1333,RGBColor[1.000,0.070,0.079],det<0.2666,RGBColor[1.000,0.369,0.067],det<0.4,RGBColor[1.000,0.681,0.049],det<0.5333,RGBColor[0.984,1.000,0.051],det<0.6666,RGBColor[0.673,1.000,0.058],det<0.8,RGBColor[0.364,1.000,0.055],det<0.9333,RGBColor[0.071,1.000,0.060],det<1.066,RGBColor[0.085,1.000,0.388],det<1.2,RGBColor[0.070,1.000,0.678],det<1.333,RGBColor[0.070,        1.000,1.000],det<1.466,RGBColor[0.067,0.698,        1.000],det<1.6,RGBColor[0.075,0.401,1.000],        det<1.733,RGBColor[0.067,0.082,        1.000],det<1.866,RGBColor[0.326,0.056,        1.000],det£2,RGBColor[0.674,0.079,1.000]];        ParametricPlot3D[{Sin [theta] Cos [phi],Sin [theta] SUN        [phi],Cos [theta],L2MXYcolors[theta,phi, 1+Sin [theta] Sin        [theta] Sin [2phi]]}, {theta,0,Pi},        {phi,0,2Pi},Boxed®False,Axes®False,Lighting®False,PlotPoints®{20,20},ViewPoint®{−0.273,−2.030,3.494}];

The radii of orbitspheres of the electrons of each excited-state atomand atomic ion are calculated by solving the force balance equationgiven by Maxwell's equations for a given set of quantum numbers, and thestate is displayed as modulated charge-density waves on eachtwo-dimensional orbitsphere at each calculated radius. A computerrendering of the helium atom in the n=2, l=1 excited state according tothe present Invention is shown in FIG. 7. The algorithm used was

<<Calculus‘VectorAnalysis’

<<Graphics‘ParametricPlot3D’

<<Graphics‘Shapes’

<<Graphics‘Animation’

<<Graphics‘SurfaceOfRevolution’

<<Graphics‘Colors’

-   Electron=SphericalPlot3D[Evaluate[Append[{0.25},{Green}]],    {theta,0,Pi},    {theta,0,2Pi-Pi/2},Boxed\[Rule]False,Axes\[Rule]False,PlotPoints\[Rule]{20,20},Lighting\[Rule]False];-   Proton=Show[Graphics3D[{Red,PointSize[0.01],Point[{0,0,0}]}],Boxed\[Rule]False];-   InnerH=Show[Electron,Proton];-   L1MXcolors[theta_, phi_, det_]=    -   Which [det<0.1333, RGBColor[1.000, 0.070, 0.079],    -   det<0.2666, RGBColor[1.000, 0.369, 0.067],    -   det<0.4, RGBColor[1.000, 0.681, 0.049],    -   det<0.5333, RGBColor[0.984, 1.000, 0.051],    -   det<0.6666, RGBColor[0.673, 1.000, 0.058],    -   det<0.8, RGBColor[0.364, 1.000, 0.055],    -   det<0.9333, RGBColor[0.071, 1.000, 0.060],    -   det<1.066, RGBColor[0.085, 1.000, 0.388],    -   det<1.2, RGBColor[0.070, 1.000, 0.678],    -   det<1.333, RGBColor[0.070, 1.000, 1.000],    -   det<1.466, RGBColor[0.067, 0.698, 1.000],    -   det<1.6, RGBColor[0.075, 0.401, 1.000],    -   det<1.733, RGBColor[0.067, 0.082, 1.000],    -   det<1.866, RGBColor[0.326, 0.056, 1.000],    -   det<=2, RGBColor[0.674, 0.079, 1.000]];-   \!\(\(Do[\[IndentingNewLine]L1MX=ParametricPlot3D[Evaluate[Append[\[IndentingNewLine]    {Sin [theta]\ Cos [phi], Sin [theta]\ Sin [phi], Cos [theta]},    \[IndentingNewLine]{EdgeForm[ ], L1MXcolors[theta, phi+\((\(\(2    Pi\)V30\)i) \), 1+Sin [theta]\ Cos [phi+\((\(\(2Pi\)V30\)i)    \)]]}\[IndentingNewLine]]], {theta, 0, Pi}, {phi, 0, Pi/2+Pi}, Boxed    \[Rule] False, Axes \[Rule] False, Lighting \[Rule] False,    PlotPoints \[Rule] {35, 35}, ViewPoint \[Rule] {0,0, 2}, ImageSize    \[Rule] 72*6], \[IndentingNewLine] {i, 1, 1}]; \)\)    Show[InnerH,L1MX,Lighting[Rule]False];

The present Invention may be embodied in other specific forms withoutdeparting from the spirit or essential attributes thereof and,accordingly, reference should be made to the appended claims, ratherthan to the foregoing specification, as indicating the scope of theInvention.

References which are incorporated herein by reference in their entiretyand referred to above throughout using [brackets]:

-   1. R. Mills, The Grand Unified Theory of Classical Quantum    Mechanics, January 2005 Edition; posted at    http://www.blacklightpower.com/bookdownload.shtml.-   2. R. L. Mills, “The Grand Unified Theory of Classical Quantum    Mechanics”, Int. J. Hydrogen Energy, Vol. 27, No. 5, (2002), pp.    565-590.-   3. R. L. Mills, “Classical Quantum Mechanics”, submitted; posted at    http://www.blacklightpower.com/pdf/CQMTheoryPaperTablesand%20Figures080403.pdf.-   4. R. L. Mills, “The Nature of the Chemical Bond Revisited and an    Alternative Maxwellian Approach”, submitted; posted at    http://www.blacklightpower.com/pdf/technical/H2    PaperTableFiguresCaptions111303.pdf.-   5. R. L. Mills, “Exact Classical Quantum Mechanical Solutions for    One-Through Twenty-Electron Atoms”, submitted; posted at    http://www.blacklightpower.com/pdf/technical/Exact%20Classical%20Quantum%20Mechanical%20Solutions%20for%20One-%20Through%20Twenty-Electron%20Atoms%20042204.pdf.-   6. R. L. Mills, “Maxwell's Equations and QED: Which is Fact and    Which is Fiction”, submitted; posted at    http://www.blacklightpower.com/pdf/technical/MaxwellianEquationsandQED080604.pdf.-   7. R. L. Mills, “Exact Classical Quantum Mechanical Solution for    Atomic Helium Which Predicts Conjugate Parameters from a Unique    Solution for the First Time”, submitted; posted at    http://www.blacklightpower.com/pdf/technical/ExactCQMSolutionforAtomicHelium073004.pdf.-   8. R. L. Mills, “The Fallacy of Feynman's Argument on the Stability    of the Hydrogen Atom According to Quantum Mechanics”, submitted;    posted    athttp://www.blacklightpower.com/pdf/Feynman%27s%20Argument%20Spec%20UPDATE%20091003.pdf.-   9. R. Mills, “The Nature of Free Electrons in Superfluid Helium—a    Test of Quantum Mechanics and a Basis to Review its Foundations and    Make a Comparison to Classical Theory”, Int. J. Hydrogen Energy,    Vol. 26, No. 10, (2001), pp. 1059-1096.-   10. R. Mills, “The Hydrogen Atom Revisited”, Int. J. of Hydrogen    Energy, Vol. 25, Issue 12, December, (2000), pp. 1171-1183.-   11. P. Pearle, Foundations of Physics, “Absence of radiationless    motions of relativistically rigid classical electron”, Vol. 7, Nos.    11/12, (1977), pp. 931-945.-   12. V. F. Weisskopf, Reviews of Modern Physics, Vol. 21, No. 2,    (1949), pp. 305-315.-   13. H. Wergeland, “The Klein Paradox Revisited”, Old and New    Questions in Physics, Cosmology, Philosophy, and Theoretical    Biology, A. van der Merwe, Editor, Plenum Press, New York, (1983),    pp. 503-515.-   14. A. Einstein, B. Podolsky, N. Rosen, Phys. Rev., Vol. 47,    (1935), p. 777.-   15. F. Dyson, “Feynman's proof of Maxwell equations”, Am. J. Phys.,    Vol. 58, (1990), pp. 209-211.-   16. H. A. Haus, “On the radiation from point charges”, American    Journal of Physics, Vol. 54, 1126-1129 (1986).-   17. http://www.blacklightpower.com/new.shtml.-   18. D. A. 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1. A system of computing and rendering the nature of excited-stateatomic and atomic ionic electrons comprising: physical, Maxwelliansolutions of the charge, mass, and current density functions of atomsand atomic ions, a processing means, and an output means.
 2. The systemof claim 1, further comprising a display in communication with theoutput means.
 3. The system of claim 2, wherein the display is at leastone of visual or graphical media.
 4. The system of claim 2, wherein thedisplay is at least one of static or dynamic.
 5. The system of claim 4,wherein at least one of spin and orbital angular motion is be displayed.6. The system of claim 2, wherein the displayed information is used tomodel at least one of reactivity, physical properties, opticalabsorption and optical emission.
 7. The system of claim 2, wherein thedisplayed information is used to model other of excited-state atoms andatomic ions and provide utility to anticipate their reactivity, physicalproperties, optical absorption and optical emission.
 8. The system ofclaim 1, wherein the processing means is a general purpose computer. 9.The system of claim 8, wherein the general purpose computer comprises acentral processing unit (CPU), one or more specialized processors,system memory, a mass storage device such as a magnetic disk, an opticaldisk, or other storage device, an input means such as a keyboard ormouse, a display device, and a printer or other output device.
 10. Thesystem of claim 1, wherein the processing means comprises a specialpurpose computer or other hardware system.
 11. The system of claim 1,further comprising computer program products.
 12. The system of claim11, comprising computer readable medium having embodied therein programcode means.
 13. The system of claim 12, wherein the computer readablemedia is any available media which can be accessed by a general purposeor special purpose computer.
 14. The system of claim 13, wherein thecomputer readable media comprises at least one of RAM, ROM, EPROM, CDROM, DVD or other optical disk storage, magnetic disk storage or othermagnetic storage devices, or any other medium which can embody thedesired program code means and which can be accessed by a generalpurpose or special purpose computer.
 15. The system of claim 14, whereinthe program code means comprises executable instructions and data whichcause a general purpose computer or special purpose computer to performa certain function of a group of functions.
 16. The system of claim 15,wherein the program code is Mathematica programmed with an algorithmbased on the physical solutions, and the computer is a PC.
 17. Thesystem of claim 16, wherein the algorithm for the rendering of theelectron of atomic hydrogen using Mathematica and computed on a PC isSphericalPlot3D[1, {q,0,p}, {f,0,2p},Boxed®False,Axes®False]; and thealgorithm for the rendering of atomic hydrogen using Mathematica andcomputed on a PC is Electron=SphericalPlot3D[1, {q,0,p},{f,0,2p-p/2},Boxed®False,Axes®False];Proton=Show[Graphics3D[{Blue,PointSize[0.03],Point[{0,0,0}]}],Boxed®False];Show [Electron,Proton].
 18. The system of claim 16, wherein thealgorithm for the rendering of thespherical-and-time-harmonic-electron-charge-density functions usingMathematica and computed on a PC are To generate L1MO:L1MOcolors[theta_,phi_,det_=Which[det<0.1333,RGBColor[1.000,0.070,0.079],det<0.2666,RGBColor[1.000,0.369,0.067],det<0.4,RGBColor[1.000,0.681,0.049],det<0.5333,RGBColor[0.984,1.000,0.051],det<0.6666,RGBColor[0.673,1.000,0.058],det<0.8,RGBColor[0.364,1.000,0.055],det<0.9333,RGBColor[0.071,1.000,0.060],det<1.066,RGBColor[0.085,1.000,0.388],det<1.2,RGBColor[0.070,1.000,0.678],det<1.333,RGBColor[0.070,1.000,1.000],det<1.466,RGBColor[0.067,0.698,1.000],det<1.6,RGBColor[0.075,0.401,1.000],det<1.733,RGBColor[0.067,0.082,1.000],det<1.866,RGBColor[0.326,0.056,1.000],det£2,RGBColor[0.674,0.079,1.000]]; L1MO=ParametricPlot3D[{Sin[theta] Cos [phi],Sin [theta] Sin [phi],Cos [theta],L1MOcolors[theta,phi,1+Cos [theta]]}, {theta,0,Pi},{phi,0,2Pi},Boxed®False,Axes®False,Lighting®False,PlotPoints®{20,20},ViewPoint®{−0.273,−2.030,3.494}];To generate L1MX: L1MXcolors[theta_, phi_, det_i=Which [det<0.1333,RGBColor[1.000, 0.070, 0.079],det<0.2666, RGBColor[1.000, 0.369,0.067],det<0.4, RGBColor[1.000, 0.681, 0.049],det<0.5333,RGBColor[0.984, 1.000, 0.051],det<0.6666, RGBColor[0.673, 1.000, 0.058],det<0.8, RGBColor[0.364, 1.000, 0.055],det<0.9333, RGBColor[0.071,1.000, 0.060], det<1.066, RGBColor[0.085, 1.000, 0.388],det<1.2,RGBColor[0.070, 1.000, 0.678], det<1.333, RGBColor[0.070, 1.000,1.000],det<1.466, RGBColor[0.067, 0.698, 1.000], det<1.6,RGBColor[0.075, 0.401, 1.000],det<1.733, RGBColor[0.067, 0.082, 1.000],det<1.866, RGBColor[0.326, 0.056, 1.000],det<=2, RGBColor[0.674, 0.079,1.000]]; L1MX=ParametricPlot3D[{Sin [theta] Cos [phi],Sin [theta] SUN[phi],Cos [theta],L1MXcolors[theta,phi, 1+Sin [theta] Cos [phi]]},{theta,0,Pi},{phi,0,2Pi},Boxed®False,Axes®False,Lighting®False,PlotPoints®{20,20},ViewPoint®{−0.273,−2.030,3.494}];To generate L1MY: L1MYcolors[theta_,phi_,det_]=Which[det<0.1333,RGBColor[1.000,0.070,0.079],det<0.2666,RGBColor[1.000,0.369,0.067],det<0.4,RGBColor[1.000,0.681,0.049],det<0.5333,RGBColor[0.984,1.000,0.051],det<0.6666,RGBColor[0.673,1.000,0.058],det<0.8,RGBColor[0.364,1.000,0.055],det<0.9333,RGBColor[0.071,1.000,0.060],det<1.066,RGBColor[0.085,1.000,0.388],det<1.2,RGBColor[0.070,1.000,0.678],det<1.333,RGBColor[0.070,1.000,1.000],det<1.466,RGBColor[0.067,0.698,1.000],det<1.6,RGBColor[0.075,0.401,1.000],det<1.733,RGBColor[0.067,0.082,1.000],det<1.866,RGBColor[0.326,0.056,1.000],det£2,RGBColor[0.674,0.079,1.000]];L1MY=ParametricPlot3D[{Sin [theta] Cos [phi],Sin [theta] SUN [phi],Cos[theta],L1 MYcolors[theta,phi, 1+Sin [theta] SUN [phi]]}, {theta,0,Pi},{phi,0,2Pi},Boxed®False,Axes®False,Lighting®False,PlotPoints®{20,20}];To generate L2MO: L2MOcolors[theta_, phi_, det_i=Which [det<0.2,RGBColor[1.000, 0.070, 0.079],det<0.4, RGBColor[1.000, 0.369,0.067],det<0.6, RGBColor[1.000, 0.681, 0.049],det<0.8, RGBColor[0.984,1.000, 0.051],det<1, RGBColor[0.673, 1.000, 0.058],det<1.2,RGBColor[0.364, 1.000, 0.055],det<1.4, RGBColor[0.071, 1.000,0.060],det<1.6, RGBColor[0.085, 1.000, 0.388],det<1.8, RGBColor[0.070,1.000, 0.678],det<2, RGBColor[0.070, 1.000, 1.000],det<2.2,RGBColor[0.067, 0.698, 1.000],det<2.4, RGBColor[0.075, 0.401,1.000],det<2.6, RGBColor[0.067, 0.082, 1.000],det<2.8, RGBColor[0.326,0.056, 1.000],det<=3, RGBColor[0.674, 0.079, 1.000]];L2MO=ParametricPlot3D[{Sin [theta] Cos [phi], Sin [theta] Sin [phi], Cos[theta], L2MOcolors[theta, phi, 3 Cos [theta] Cos [theta]]}, {theta, 0,Pi}, {phi, 0, 2Pi}, Boxed->False, Axes->False, Lighting->False,PlotPoints->{20, 20}, ViewPoint->{−0.273, −2.030, 3.494}]; To generateL2MF: L2MFcolors[theta_,phi_,det_]=Which[det<0.1333,RGBColor[1.000,0.070,0.079],det<0.2666,RGBColor[1.000,0.369,0.067],det<0.4,RGBColor[1.000,0.681,0.049],det<0.5333,RGBColor[0.984,1.000,0.051],det<0.6666,RGBColor[0.673,1.000,0.058],det<0.8,RGBColor[0.364,1.000,0.055],det<0.9333,RGBColor[0.071,1.000,0,060],det<1.066,RGBColor[0.085,1.000,0.388],det<1.2,RGBColor[0.070,1.000,0.678],det<1.333,RGBColor[0.070,1.000,1.000],det<1.466,RGBColor[0.067,0.698,1.000],det<1.6,RGBColor[0.075,0.401,1.000],det<1.733,RGBColor[0.067,0.082,1.000],det<1.866,RGBColor[0.326,0.056,1.000],det£2,RGBColor[0.674,0.079,1.000]]; L2MF=ParametricPlot3D[{Sin [theta] Cos [phi],Sin [theta] SUN[phi],Cos [theta],L2MFcolors[theta,phi,1+0.72618 Sin [theta] Cos [phi] 5Cos [theta] Cos [theta]-0.72618 Sin [theta] Cos [phi]]}, {theta,0,Pi},{phi,0,2Pi},Boxed®False,Axes®False,Lighting®False,PlotPoints®{20,20},ViewPoint®{−0.273,−2.030,2.494}];To generate L2MX2Y2: L2MX2Y2colors[theta_,phi_,det_]=Which[det<0.1333,RGBColor[1.000,0.070,0.079],det<0.2666,RGBColor[1.000,0.369,0.067],det<0.4,RGBColor[1.000,0.681,0.049],det<0.5333,RGBColor[0.984,1.000,0.051],det<0.6666,RGBColor[0.673,1.000,0.058],det<0.8,RGBColor[0.364,1.000,0.055],det<0.9333,RGBColor[0.071,1.000,0.060],det<1.066,RGBColor[0.085,1.000,0.388],det<1.2,RGBColor[0.070,1.000,0.678],det<1.333,RGBColor[0.070,1.000,1.000],det<1.466,RGBColor[0.067,0.698,1.000],det<1.6,RGBColor[0.075,0.401,1.000],det<1.733,RGBColor[0.067,0.082,1.000],det<1.866,RGBColor[0.326,0.056,1.000],det£2,RGBColor[0.674,0.079, 1.000]];L2MX2Y2=ParametricPlot3D[{Sin [theta] Cos [phi],Sin [theta] SUN[phi],Cos [theta],L2MX2Y2colors[theta,phi,1+Sin [theta] Sin [theta] Cos[2 phi]]}, {theta,0,Pi},{phi,0,2Pi},Boxed®False,Axes®False,Lighting®False,PlotPoints®{20,20},ViewPoint®{−0.273,−2.030,3.494}];To generate L2MXY: L2MXYcolors[theta_,phi_,det_=Which[det<0.1333,RGBColor[1.000,0.070,0.079],det<0.2666,RGBColor[1.000,0.369,0.067],det<0.4,RGBColor[1.000,0.681,0.049],det<0.5333,RGBColor[0.984,1.000,0.051],det<0.6666,RGBColor[0.673,1.000,0.058],det<0.8,RGBColor[0.364,1.000,0.055],det<0.9333,RGBColor[0.071,1.000,0.060],det<1.066,RGBColor[0.085,1.000,0.388],det<1.2,RGBColor[0.070,1.000,0.678],det<1.333,RGBColor[0.070,1.000,1.000],det<1.466,RGBColor[0.067,0.698,1.000],det<1.6,RGBColor[0.075,0.401,1.000],det<1.733,RGBColor[0.067,0.082,1.000],det<1.866,RGBColor[0.326,0.056,1.000],det£2,RGBColor[0.674,0.079,1.000]]; ParametricPlot3D[{Sin [theta] Cos [phi],Sin [theta] SUN[pbi],Cos [theta],L2MXYcolors[theta,phi,1+Sin [theta] Sin [theta] Sin [2phi]]}, {theta,0,Pi},{phi,0,2Pi},Boxed®False,Axes®False,Lighting®False,PlotPoints®{20,20},ViewPointOR{−0.273,−2.030,3.494}].
 19. The system of claim 16, wherein thealgorithm for the rendering of thespherical-and-time-harmonic-electron-charge-density functions usingMathematica and computed on a PC for the helium atom in the n=2, l=1excited state is <<Calculus‘VectorAnalysis’ <<Graphics‘ParametricPlot3D’<<Graphics‘Shapes’ <<Graphics‘Animation’ <<Graphics‘SurfaceOfRevolution’<<Graphics‘Colors’ Electron=SphericalPlot3D[Evaluate[Append[{0.25},{Green}]], {theta,0,Pi}, {theta,0,2Pi-Pi/2},Boxed\[Rule]False,Axes\[Rule]False,PlotPoints\[Rule] {20,20},Lighting\[Rule]False];Proton=Show[Graphics3D[{Red,PointSize[0.0],Point[{0,0,0}]}],Boxed[Rule]False];InnerH=Show[Electron,Proton]; L1MXcolors[theta_, phi_, det_]= Which[det<0.1333, RGBColor[1.000, 0.070, 0.079], det<0.2666, RGBColor[1.000,0.369, 0.067], det<0.4, RGBColor[1.000, 0.681, 0.049], det<0.5333,RGBColor[0.984, 1.000, 0.051], det<0.6666, RGBColor[0.673, 1.000,0.058], det<0.8, RGBColor[0.364, 1.000, 0.055], det<0.9333,RGBColor[0.071, 1.000, 0.060], det<1.066, RGBColor[0.085, 1.000, 0.388],det<1.2, RGBColor[0.070, 1.000, 0.678], det<1.333, RGBColor[0.070,1.000, 1.000], det<1.466, RGBColor[0.067, 0.698, 1.000], det<1.6,RGBColor[0.075, 0.401, 1.000], det<1.733, RGBColor[0.067, 0.082, 1.000],det<1.866, RGBColor[0.326, 0.056, 1.000], det<=2, RGBColor[0.674, 0.079,1.000]]; \!\(\(Do[\[IndentingNewLine]L1MX=ParametricPlot3D[Evaluate[Append[\[IndentingNewLine] {Sin [theta]\Cos [phi], Sin [theta]\ Sin [phi], Cos [theta]},\[IndentingNewLine]{EdgeForm[ ], L1MXcolors[theta, phi+\((\(\(2Pi\)V30\)i)\), 1+Sin [theta]\ Cos [phi+\((\(\(2 Pi\)V30\)i)\)]]}\[IndentingNewLine]]], {theta, 0, Pi}, {phi, 0, Pi/2+Pi}, Boxed\[Rule] False, Axes \[Rule] False, Lighting \[Rule] False, PlotPoints\[Rule] {35, 35}, ViewPoint \[Rule] {0, 0, 2}, ImageSize \[Rule] 72*6],\[IndentingNewLine] {i, 1, 1}];\)\)Show[InnerH,L1MX,Lighting\[Rule]False].
 20. The system of claim 1,wherein the physical, Maxwellian solutions of the charge, mass, andcurrent density functions of excited-state atoms and atomic ionscomprises a solution of the classical wave equation${\left\lbrack {{\nabla^{2}{- \frac{1}{v^{2}}}}\frac{\partial^{2}}{\partial t^{2}}} \right\rbrack{\rho\left( {r,\theta,\phi,t} \right)}} = 0.$21. The system of claim 20, wherein the time, radial, and angularsolutions of the wave equation are separable.
 22. The system of claim21, wherein the radii of orbitspheres of the electrons of eachexcited-state atom and atomic ion are calculated by solving the forcebalance equation given by Maxwell's equation for a given set of quantumnumbers, and the state is displayed as modulated charge-density waves oneach two-dimensional orbitsphere at each calculated radius.
 23. Thesystem of claim 21, wherein radial function which does satisfy theboundary conditions is a radial delta function${f(r)} = {\frac{1}{r^{2}}{{\delta\left( {r - r_{n}} \right)}.}}$ 24.The system of claim 23, wherein the boundary condition is met for a timeharmonic function when the relationship between an allowed radius andthe electron wavelength is given by${{2\pi\quad r_{n}} = \lambda_{n}},{\omega = \frac{\hslash}{m_{e}r^{2}}},\quad{and}$$v = \frac{\hslash}{m_{e}r}$ where ω is the angular velocity of eachpoint on the electron surface, ν is the velocity of each point on theelectron surface, and r is the radius of the electron.
 25. The system ofclaim 24, wherein the spin function is given by the uniform function Y₀⁰(φ,θ) comprising angular momentum components of$L_{xy} = \frac{\hslash}{4}$ and $L_{z} = {\frac{\hslash}{2}.}$
 26. Thesystem of claim 25, wherein the atomic and atomic ionic charge andcurrent density functions of excited-state electrons are described by acharge-density (mass-density) function which is the product of a radialdelta function, two angular functions (spherical harmonic functions),and a time harmonic function:${{\rho\left( {r,\theta,\phi,t} \right)} = {{{f(r)}{A\left( {\theta,\phi,t} \right)}} = {\frac{1}{r^{2}}{\delta\left( {r - r_{n}} \right)}{A\left( {\theta,\phi,t} \right)}}}};$A(θ, ϕ, t) = Y(θ, ϕ)k(t) wherein the spherical harmonic functionscorrespond to a traveling charge density wave confined to the sphericalshell which gives rise to the phenomenon of orbital angular momentum.27. The system of claim 26, wherein based on the radial solution, theangular charge and current-density functions of the electron, A(θ,φ,t),must be a solution of the wave equation in two dimensions (plus time),${\left\lbrack {{\nabla^{2}{- \frac{1}{v^{2}}}}\frac{\partial^{2}}{\partial t^{2}}} \right\rbrack{A\left( {\theta,\phi,t} \right)}} = 0$where ρ(r,θ,φ,t)=ƒ(r)A(θ,φt)=1/r²δ(r−r_(n))A(θ,φ,t) andA(θ,φ,t)=Y(θ,φ)k(t)${\left\lbrack {{\frac{1}{r^{2}\sin\quad\theta}\frac{\partial}{\partial\theta}\left( {\sin\quad\theta\frac{\partial}{\partial\theta}} \right)_{r,\phi}} + {\frac{1}{r^{2}\sin^{2}\theta}\left( \frac{\partial^{2}}{\partial\phi^{2}} \right)_{r,\theta}} - {\frac{1}{v^{2}}\frac{\partial^{2}}{\partial t^{2}}}} \right\rbrack{A\left( {\theta,\phi,t} \right)}} = 0$where ν is the linear velocity of the electron.
 28. The system of claim27, wherein the charge-density functions including the time-functionfactor are  = 0 ρ ⁡ ( r , θ , ϕ , t ) = e 8 ⁢ π ⁢   ⁢ r 2 ⁡ [ δ ⁡ ( r - r n )] ⁡ [ Y 0 0 ⁡ ( θ , ϕ ) + Y m ⁡ ( θ , ϕ ) ]  ≠ 0 ρ ⁡ ( r , θ , ϕ , t ) = e 4⁢π ⁢   ⁢ r 2 ⁡ [ δ ⁡ ( r - r n ) ] ⁡ [ Y 0 0 ⁡ ( θ , ϕ ) + Re ⁢ { Y m ⁡ ( θ , ϕ )⁢ⅇ ⅈω n ⁢ t } ] where Y_(l) ^(m)(θ,φ) are the spherical harmonic functionsthat spin about the z-axis with angular frequency ω_(n) with Y₀ ⁰(θ,φ)the constant function; Re {Y_(l) ^(m)(θφ)e^(iω) ^(n) ^(j)}=P_(l)^(m)(cos θ)cos(mφ+{dot over (ω)}_(n)t) where to keep the form of thespherical harmonic as a traveling wave about the z-axis, {dot over(ω)}_(n)=mω_(n).
 29. The system of claim 28, wherein the spin andangular moment of inertia, I, angular momentum, L, and energy, E, forquantum number l are given by  = 0$I_{z} = {I_{spin} = \frac{m_{e}r_{n}^{2}}{2}}$$L_{z} = {{I\quad\omega\quad i_{z}} = {\pm \frac{\hslash}{2}}}$$\begin{matrix}{E_{rotational} = E_{{rotational},{spin}}} \\{= {\frac{1}{2}\left\lbrack {I_{spin}\left( \frac{\hslash}{m_{e}r_{n}^{2}} \right)}^{2} \right\rbrack}} \\{= {\frac{1}{2}\left\lbrack {\frac{m_{e}r_{n}^{2}}{2}\left( \frac{\hslash}{m_{e}r_{n}^{2}} \right)^{2}} \right\rbrack}} \\{= {\frac{1}{4}\left\lbrack \frac{\hslash^{2}}{2I_{spin}} \right\rbrack}}\end{matrix}$  ≠ 0 I orbital = m e ⁢ r n 2 ⁡ [ ⁢ ( + 1 ) 2 + + 1 ] 1 2L_(z) = m  ℏ L_(z  total) = L_(z  spin) + L_(z  orbital) E rotional ,orbital = ℏ 2 2 ⁢ I ⁡ [ ⁢ ( + 1 ) 2 + 2 ⁢ + 1 ]$T = \frac{\hslash^{2}}{2m_{e}r_{n}^{2}}$ ⟨E_(rotational, orbital)⟩ = 0.30. The system of claim 1, wherein the initial force balance equationfor one-electron atoms and ions before excitation is${\frac{m_{e}}{4\pi\quad r_{1}^{2}}\frac{v_{1}^{2}}{r_{1}}} = {{\frac{e}{4\pi\quad r_{1}^{2}}\frac{Z\quad e}{4{\pi ɛ}_{o}r_{1}^{2}}} - {\frac{1}{4\pi\quad r_{1}^{2}}\frac{\hslash^{2}}{m_{p}r_{n}^{2}}}}$$r_{1} = \frac{a_{H}}{Z}$ where α_(H) is the radius of the hydrogenatom.
 31. The system of claim 30, wherein from Maxwell's equations, thepotential energy V, kinetic energy T, electric energy or binding energyE_(ele) are $\begin{matrix}{V = {\frac{{- Z}\quad{\mathbb{e}}^{2}}{4{\pi ɛ}_{o}r_{1}} = \frac{{- Z^{2}}\quad{\mathbb{e}}^{2}}{4{\pi ɛ}_{o}a_{H}}}} \\{= {{{- Z^{2}}X\quad 4.3675\quad X\quad 10^{- 18}J} = {{- Z^{2}}X\quad 27.2\quad e\quad V}}}\end{matrix}$$T = {\frac{Z^{2}{\mathbb{e}}^{2}}{8{\pi ɛ}_{o}a_{H}} = {Z^{2}X\quad 13.59\quad e\quad V}}$$T = {E_{ele} = {{{- \frac{1}{2}}ɛ_{o}{\int_{\infty}^{r_{1}}{E^{2}{\mathbb{d}v}\quad{where}\quad E}}} = {- \frac{Z\quad e}{4{\pi ɛ}_{o}r^{2}}}}}$$E_{ele} = {{- \frac{Z^{2}\quad{\mathbb{e}}^{2}}{8{\pi ɛ}_{o}a_{H}}} = {{{- Z^{2}}X\quad 2.1786\quad X\quad 10^{- 18}\quad J} = {{- Z^{2}}X\quad 13.598\quad e\quad{V.}}}}$32. The system of claim 1, wherein the initial force balance equationsolution before excitation of two-electron atoms is a central forcebalance equation with the nonradiation condition given by${\frac{m_{e}}{4\pi\quad r_{2}^{2}}\frac{v_{2}^{2}}{r_{2}}} = {{\frac{e}{4\pi\quad r_{2}^{2}}\frac{\left( {Z - 1} \right)\quad e}{4{\pi ɛ}_{o}r_{2}^{2}}} + {\frac{1}{4\pi\quad r_{2}^{2}}\frac{\hslash^{2}}{Z\quad m_{e}r_{2}^{3}}\sqrt{s\left( {s + 1} \right)}}}$which gives the radius of both electrons as${r_{2} = {r_{1} = {a_{0}\left( {\frac{1}{Z - 1} - \sqrt{\frac{s\left( {s + 1} \right)}{Z\left( {Z - 1} \right)}}} \right)}}};\quad{s = {\frac{1}{2}.}}$33. The system of claim 32, wherein the ionization energy for helium,which has no electric field beyond r, is given byIonization Energy(He)=−E(electric)+E(magnetic) where,${E({electric})} = {- \frac{\left( {Z - 1} \right){\mathbb{e}}^{2}}{8{\pi ɛ}_{o}r_{1}}}$${E({magnetic})} = \frac{2{\pi\mu}_{0}{\mathbb{e}}^{2\quad}\hslash^{2}}{m_{e}^{2}r_{1}^{3}}$For 3≦Z${{Ionization}\quad{Energy}} = {{{- {Electric}}\quad{Energy}} - {\frac{1}{Z}{Magnetic}\quad{{Energy}.}}}$34. The system of claim 1, wherein the electrons of excited states ofone and multielectron atoms all exist as orbitspheres of discrete radiiwhich are given by r_(n) of the radial Dirac delta function, δ(r−r_(n)).35. The system of claim 34, wherein the electrons of excited states ofone and multielectron atoms all exist as orbitspheres of discrete radiiwhich are given by r_(n) of the radial Dirac delta function, δ(r−r_(n))that serve as resonator cavities and trap electromagnetic radiation ofdiscrete resonant frequencies.
 36. The system of claim 35, whereinphoton absorption occurs as an excitation of a resonator mode.
 37. Thesystem of claim 36, wherein the free space photon also comprises aradial Dirac delta function, and the angular momentum of the photongiven by$m = {{\int{\frac{1}{8\pi\quad c}{{Re}\left\lbrack {r \times \left( {E \times B^{*}} \right)} \right\rbrack}{\mathbb{d}x^{4}}}} = \hslash}$is conserved for the solutions for the resonant photons and excitedstate electron functions.
 38. The system of claim 37, wherein the changein angular frequency of the electron is equal to the angular frequencyof the resonant photon that excites the resonator cavity modecorresponding to the transition, and the energy is given by Planck'sequation.
 39. The system of claim 38, wherein for each multipole statewith a single m value the relationship between the angular momentumM_(z), energy U, and angular frequency ω is given by:$\frac{\mathbb{d}M_{z}}{\mathbb{d}r} = {\frac{m}{\omega}\frac{\mathbb{d}U}{\mathbb{d}r}}$independent of r where m is an integer such that the ratio of the squareof the angular momentum, M², to the square of the energy, U², for a pure(l, m) multipole is given by$\frac{M^{2}}{U^{2}} = {\frac{m^{2}}{\omega^{2}}.}$
 40. The system ofclaim 39, wherein the radiation from such a multipole of order (l, m)carries off m

units of the z component of angular momentum per photon of energy

ω.
 41. The system of claim 40, wherein the photon and the electron caneach posses only

of angular momentum which requires that the radiation field contain mphotons.
 42. The system of claim 41, wherein during excitation the spin,orbital, or total angular momentum of the orbitsphere can change by zeroor ±

.
 43. The system of claim 42, wherein the selection rules for multipoletransitions between quantum states arise from conservation of thephoton's multipole moment and angular momentum of

.
 44. The system of claim 43, wherein in an excited state, thetime-averaged mechanical angular momentum and rotational energyassociated with the traveling charge-density wave on the orbitsphere iszero, and the angular momentum of

of the photon that excites the electronic state is carried by the fieldsof the trapped photon.
 45. The system of claim 44, wherein theamplitudes of the rotational moment of inertia, angular momentum, andenergy that couple to external magnetic and electromagnetic fields aregiven by ⁢   ⁢ I orbital = m e ⁢ r n 2 ⁡ [ ⁢ ( + 1 ) 2 + 2 ⁢ + 1 ] 1 2 = m e ⁢r n 2 ⁢ + 1 , ⁢ L = I ⁢   ⁢ ω ⁢   ⁢ i z = I orbital ⁢ ω ⁢   ⁢ i z = m e ⁢ r n 2 ⁡ [⁢( + 1 ) 2 + 2 ⁢ + 1 ] 1 2 ⁢ ω ⁢   ⁢ i z = m e ⁢ r n 2 ⁢ ℏ m e ⁢ r n 2 ⁢ + 1 = ℏ⁢ + 1 , and E rotational ⁢   ⁢ orvital ⁢ = ℏ 2 2 ⁢ I ⁡ [ ⁢ ( + 1 ) 2 + 2 ⁢ + 1 ]= ℏ 2 2 ⁢ I ⁡ [ + 1 ] = ℏ 2 2 ⁢ m e ⁢ r n 2 ⁡ [ + 1 ] , respectively . 46.The system of claim 45, wherein the electron charge-density waves arenonradiative due to the angular motion, but excited states are radiativedue to a radial dipole that arises from the presence of the trappedphoton.
 47. The system of claim 46, wherein the total number ofmultipoles, N_(l,s), of an energy level corresponding to a principalquantum number n where each multipole corresponds to an l and ml quantumnumber is N , s = ∑ = 0 n - 1 ⁢ ∑   ⁢ m = - + ⁢ 1 = ∑ = 0 n - 1 ⁢ 2 ⁢ + 1 =( + 1 ) 2 = 2 + 2 ⁢ + 1 = n 2 .
 48. The system of claim 47, wherein thephoton's electric field superposes that of the nucleus for r₁<r<r₂ suchthat the radial electric field has a magnitude proportional to e/n atthe electron 2 (the excited electron) where n=2, 3, 4, . . . for excitedstates such that U is decreased by the factor of 1/n².
 49. The system ofclaim 48, wherein the “trapped photon” of the excited state is a“standing electromagnetic wave” which actually is a traveling wave thatpropagates on the surface around the z-axis, and its source current isonly at the orbitsphere.
 50. The system of claim 49, wherein thetime-function factor, k(t), for the “standing wave” is identical to thetime-function factor of the orbitsphere in order to satisfy the boundary(phase) condition at the orbitsphere surface such that the angularfrequency of the “trapped photon” is identical to the angular frequencyof the electron orbitsphere, ω_(n).
 51. The system of claim 50, whereinthe angular functions of the “trapped photon” are identical to thespherical harmonic angular functions of the electron orbitsphere. 52.The system of claim 51, wherein combining k(t) with the φ-functionfactor of the spherical harmonic gives e^(i(mφ−ω) ^(n) ^(t)) for boththe electron and the “trapped photon” function.
 53. The system of claim52, wherein the photon “standing wave” in an excited electronic state isa solution of Laplace's equation in spherical coordinates with sourcecurrents matching those of the electron orbitsphere “glued” to theelectron and phase-locked to the electron current density wave thattravel on the surface with a radial electric field.
 54. The system ofclaim 53, wherein the photon field is purely radial since the field istraveling azimuthally at the speed of light even though the sphericalharmonic function has a velocity less than light speed.
 55. The systemof claim 54, wherein the photon field does not change the nature of theelectrostatic field of the nucleus or its energy except at the positionof the electron.
 56. The system of claim 55, wherein the photon“standing wave” function comprises a radial Dirac delta function that“samples” the Laplace equation solution only at the positioninfinitesimally inside of the electron current-density function andsuperimposes with the proton field to give a field of radial magnitudecorresponding to a charge of e/n where n=2, 3, 4, . . . .
 57. The systemof claim 56, wherein the electric field of the nucleus for r₁<r<r₂ is$E_{nucleus} = {\frac{e}{4{\pi ɛ}_{o}r^{2}}.}$
 58. The system of claim57, wherein the equation of the electric field of the “trapped photon”for r=r₂ where r₂ is the radius of electron 2, is E r ⁢   ⁢ photon ⁢   ⁢ n ,l , m ⁢ | r = r 2 = e 4 ⁢ πɛ o ⁢ r 2 2 ⁡ [ - 1 + 1 n ⁡ [ Y 0 0 ⁡ ( θ , ϕ ) +Re ⁢ { Y m ⁡ ( θ , ϕ ) ⁢ ⅇ ⅈω n ⁢ t } ] ] ⁢ δ ⁡ ( r - r n )ω_(n) = 0  for  m =
 0. ω_(n) = 0
 59. The system of claim 58, wherein thetotal central field for r=r₂ is given by the sum of the electric fieldof the nucleus and the electric field of the “trapped photon”:E _(total) =E _(nucleus) +E _(photon).
 60. The system of claim 59,wherein for r₁<r<r₂, E r total = ⁢ e 4 ⁢ πɛ o ⁢ r 1 2 + e 4 ⁢ πɛ o ⁢ r 2 2 ⁢[ - 1 + 1 n ⁡ [ Y 0 0 ⁡ ( θ , ϕ ) + Re ⁢ { Y m ⁡ ( θ , ϕ ) ⁢ ⅇ ⅈω n ⁢ t } ] ] ⁢δ ⁡ ( r - r n ) = ⁢ 1 n ⁢ e 4 ⁢ πɛ o ⁢ r 2 2 ⁡ [ Y 0 0 ⁡ ( θ , ϕ ) + Re ⁢ { Y m ⁡( θ , ϕ ) ⁢ ⅇ ⅈω n ⁢ t } ] ⁢ δ ⁡ ( r - r n ) ω_(n) = 0  for  m =
 0. 61. Thesystem of claim 60, wherein for r=r₂ and m=0, the total radial electricfield is $E_{rtotal} = {\frac{1}{n}{\frac{e}{4{\pi ɛ}_{o}r^{2}}.}}$ 62.The system of claim 61, wherein the result is the same for the excitedstates of the one-electron atom in that the total radial electric fieldis $E_{rtotal} = {\frac{1}{n}{\frac{e}{4{\pi ɛ}_{o}r^{2}}.}}$
 63. Thesystem of claim 61, wherein for r₁<r<r₂, E r total = ⁢ e 4 ⁢ πɛ o ⁢ r 1 2 +e 4 ⁢ πɛ o ⁢ r 2 2 ⁢ [ - 1 + 1 n ⁡ [ Y 0 0 ⁡ ( θ , ϕ ) + Re ⁢ { Y m ⁡ ( θ , ϕ )⁢ⅇ ⅈω n ⁢ t } ] ] ⁢ δ ⁡ ( r - r n ) = ⁢ 1 n ⁢ e 4 ⁢ πɛ o ⁢ r 2 2 ⁡ [ Y 0 0 ⁡ ( θ ,ϕ ) + Re ⁢ { Y m ⁡ ( θ , ϕ ) ⁢ ⅇ ⅈω n ⁢ t } ] ⁢ δ ⁡ ( r - r n )ω_(n) = 0  for  m = 0
 64. The system of claim 63, wherein the radii ofthe excited-state electron is determined from the force balance of theelectric, magnetic, and centrifugal forces that corresponds to theminimum of energy of the system.
 65. The system of claim 64, wherein theexcited-state energies are given by the electric energies at theseradii.
 66. The system of claim 65, wherein electron orbitspheres may bespin paired or unpaired depending on the force balance which applies toeach electron wherein the electron configuration is a minimum of energy.67. The system of claim 66, wherein the minimum energy configurationsare given by solutions to Laplace's equation.
 68. The system of claim67, wherein the corresponding force balance of the central Coulombic,paramagnetic, and diamagnetic forces is derived for each n-electron atomthat is solved for the radius of each electron.
 69. The system of claim68, wherein the central Coulombic force is that of a point charge at theorigin since the electron charge-density functions are sphericallysymmetrical with a harmonic time dependence.
 70. The system of claim 69,wherein the ionization energy of each electron is obtained using thecalculated radii in the determination of the Coulombic and any magneticenergies.
 71. The system of claim 69, wherein for the singlet-excitedstate with l=0, the electron source current in the excited state is aconstant function that spins as a globe about the z-axis: ρ ⁡ ( r , θ , ϕ, t ) = e 8 ⁢ π ⁢   ⁢ r 2 ⁡ [ δ ⁡ ( r - r n ) ] ⁡ [ Y 0 0 ⁡ ( θ , ϕ ) + Y m ⁡ (θ , ϕ ) ] .
 72. The system of claim 70, wherein the balance between thecentrifugal and electric and magnetic forces is given by:${\frac{m_{e}v^{2}}{r_{2}} = {\frac{\hslash^{2}}{m_{e}r_{2}^{3}} = {{\frac{1}{n}\frac{{\mathbb{e}}^{2}}{4{\pi ɛ}_{o}r_{2}^{2}}} + {\frac{2}{3}\frac{1}{n}\frac{\hslash^{2}}{2m_{e}r_{2}^{3}}\sqrt{s\left( {s + 1} \right)}}}}};{s = {\frac{1}{2}.}}$73. The system of claim 71, wherein$r_{2} = {\left\lbrack {n - \frac{\sqrt{\frac{3}{4}}}{3}} \right\rbrack\alpha_{He}}$n = 2, 3, 4, …
 74. The system of claim 72, wherein the excited-stateenergy is the energy stored in the electric field, E_(ele), which is theenergy of the excited-state electron (electron 2) relative to theionized electron at rest having zero energy:$E_{ele} = {{- \frac{1}{n}}{\frac{{\mathbb{e}}^{2}}{8{\pi ɛ}_{o}r_{2}}.}}$75. The system of claim 73, wherein the forces on electron 2 due to thenucleus and electron 1 are radial/central, invariant of r₁, andindependent of r₁ with the condition that r₁<r₂, such that r₂ can bedetermined without knowledge of r₁.
 76. The system of claim 74, whereinr₁ is solved using the equal and opposite magnetic force of electron 2on electron 1 at the radius r₂ determined from the force balanceequation for electron 2 and the central Coulombic force corresponding tothe charge of the nucleus.
 77. The system of claim 75, wherein the forcebalance between the centrifugal and electric and magnetic forces is${\frac{m_{e}v^{2}}{r_{1}} = {\frac{\hslash^{2}}{m_{e}r_{1}^{3}} = {\frac{2{\mathbb{e}}^{2}}{4{\pi ɛ}_{o}r_{1}^{2}} - {\frac{1}{3n}\frac{\hslash^{2}}{m_{e}r_{2}^{3}}\sqrt{s\left( {s + 1} \right)}}}}};{s = {\frac{1}{2}.}}$such that with ${s = \frac{1}{2}},$${r_{1}^{3} - {\left( {\frac{12n}{\sqrt{3}}r_{2}^{3}} \right)r_{1}} + {\frac{6n}{\sqrt{3}}r_{2}^{3}}} = 0$n = 2, 3, 4, …$r_{1} = {r_{13} = {{- \sqrt{\frac{2}{3}g}}\left( {{\cos\frac{\theta}{3}} - {\sqrt{3}\sin\frac{\theta}{3}}} \right)}}$where r₁ and r₂ are in units of α_(He).
 78. The system of claim 76,wherein for the triplet-excited state with l≠0, time-independentcharge-density waves corresponding to the source currents travel on thesurface of the orbitsphere of electron 2 about the z-axis.
 79. Thesystem of claim 77, wherein in the triplet state, the spin-spin force isparamagnetic and twice that of the singlet state.
 80. The system ofclaim 78, wherein the force balance between the centrifugal and electricand magnetic forces is:$\frac{m_{e}v^{2^{\prime}}}{r_{2}} = {\frac{\hslash^{2}}{m_{e}r_{2}^{3}} = {{\frac{1}{n}\frac{{\mathbb{e}}^{2}}{4{\pi ɛ}_{o}r_{2}^{2}}} + {2\frac{2}{3}\frac{1}{n}\frac{\hslash^{2}}{2m_{e}r_{2}^{3}}{\sqrt{s\left( {s + 1} \right)}.}}}}$81. The system of claim 79, wherein$r_{2} = {\left\lbrack {n - \frac{\sqrt[2]{\frac{3}{4}}}{3}} \right\rbrack\alpha_{He}}$n = 2, 3, 4, …
 82. The system of claim 80, wherein the excited-stateenergy is the energy stored in the electric field, E_(ele), which is theenergy of the excited-state electron (electron 2) relative to theionized electron at rest having zero energy:$E_{ele} = {{- \frac{1}{n}}{\frac{{\mathbb{e}}^{2}}{8{\pi ɛ}_{o}r_{2}}.}}$83. The system of claim 81, wherein using r₂, r₁ is be solved using theequal and opposite magnetic force of electron 2 on electron 1 and thecentral Coulombic force corresponding to the nuclear charge such thatthe force balance between the centrifugal and electric and magneticforces is$\frac{m_{e}v^{2}}{r_{1}} = {\frac{\hslash^{2}}{m_{e}r_{1}^{3}} = {\frac{2{\mathbb{e}}^{2}}{4{\pi ɛ}_{o}r_{1}^{2}} - {\frac{2}{3n}\frac{\hslash^{2}}{m_{e}r_{2}^{3}}\sqrt{s\left( {s + 1} \right)}}}}$and with ${s = \frac{1}{2}},$${r_{1}^{3} - {\left( {\frac{6n}{\sqrt{3}}r_{2}^{3}} \right)r_{1}} + {\frac{3n}{\sqrt{3}}r_{2}^{3}}} = 0$n = 2, 3, 4, …$r_{1} = {r_{13} = {{- \sqrt{\frac{2}{3}}}{g\left( {{\cos\frac{\theta}{3}} - {\sqrt{3}\sin\frac{\theta}{3}}} \right)}}}$where r₁ and r₂ are in units of α_(He).
 84. The system of claim 82,wherein for the singlet-excited state with l≠0, the electron sourcecurrent in the excited state is the sum of constant and time-dependentfunctions where the latter, given by${\rho\left( {r,\theta,\phi,t} \right)} = {{\frac{e}{4\pi\quad r^{2}}\left\lbrack {\delta\left( {r - r_{n}} \right)} \right\rbrack}\left\lbrack {{Y_{0}^{0}\left( {\theta,\phi} \right)} + {\text{Re}\left\{ {{Y_{\ell}^{m}\left( {\theta,\phi} \right)}{\mathbb{e}}^{{\mathbb{i}\omega}_{n}t}} \right\}}} \right\rbrack}$that travels about the z-axis.
 85. The system of claim 83, wherein thecurrent due to the time dependent term corresponding to p, d, f, etc.orbitals is $\begin{matrix}{J = {\frac{\omega_{n}}{2\pi}\frac{e}{4\pi\quad r_{n}^{2}}{N\left\lbrack {\delta\left( {r - r_{n}} \right)} \right\rbrack}\text{Re}{\left\{ {Y_{\ell}^{m}\left( {\theta,\phi} \right)} \right\}\left\lbrack {{u(t)} \times r} \right\rbrack}}} \\{= {\frac{\omega_{n}}{2\pi}\frac{e}{4\pi\quad r_{n}^{2}}{N^{\prime}\left\lbrack {\delta\left( {r - r_{n}} \right)} \right\rbrack}{\left( {{P_{\ell}^{m}\left( {\cos\quad\theta} \right)}{\cos\left( {{m\quad\phi} + {\omega_{n}^{\prime}t}} \right)}} \right)\left\lbrack {u \times r} \right\rbrack}}} \\{= {\frac{\omega_{n}}{2\pi}\frac{e}{4\pi\quad r_{n}^{2}}{N^{\prime}\left\lbrack {\delta\left( {r - r_{n}} \right)} \right\rbrack}\left( {{P_{\ell}^{m}\left( {\cos\quad\theta} \right)}{\cos\left( {{m\quad\phi} + {\omega_{n}^{\prime}t}} \right)}} \right)\sin\quad\theta\quad\hat{\phi}}}\end{matrix}$ where to keep the form of the spherical harmonic as atraveling wave about the z-axis, {dot over (ω)}_(n)=mω_(n) and N and N′are normalization constants; the vectors are defined as${\hat{\phi} = {\frac{\hat{u} \times \hat{r}}{{\hat{u} \times \hat{r}}} = \frac{\hat{u} \times \hat{r}}{\sin\quad\theta}}};$û = ẑ = orbital  axis θ̂ = ϕ̂ × r̂ “ˆ” denotes the unit vectors${\hat{u} \equiv \frac{u}{u}},$ non-unit vectors are designed in bold,and the current function is normalized.
 86. The system of claim 84,wherein the general multipole field solution to Maxwell's equations in asource-free region of empty space with the assumption of a timedependence e^(iω) ^(n) ^(j) (cgs units) is$B = {\sum\limits_{\ell,m}\left\lbrack {{{a_{E}\left( {\ell,m} \right)}{f_{\ell}({kr})}X_{\ell,m}} - {\frac{i}{k}{a_{M}\left( {\ell,m} \right)}{\nabla{\times {g_{\ell}({kr})}X_{\ell,m}}}}} \right\rbrack}$${E = {\sum\limits_{\ell,m}\left\lbrack {{\frac{i}{k}{a_{E}\left( {\ell,m} \right)}{\nabla{\times {f_{\ell}({kr})}X_{\ell,m}}}} + {{a_{M}\left( {\ell,m} \right)}{g_{\ell}({kr})}X_{\ell,m}}} \right\rbrack}};$the  radial  functions  f_(ℓ)(kr)  and  g_(ℓ)(kr)  are  of  the  form:g_(ℓ)(kr) = A_(ℓ)⁽¹⁾h_(ℓ)⁽¹⁾ + A_(ℓ)⁽²⁾h_(ℓ)⁽²⁾,  andX_(ℓ, m)  is  the  vector  spherical  harmonic  defined  by${X_{\ell,m}\left( {\theta,\phi} \right)} = {\frac{1}{\sqrt{\ell\left( {\ell + 1} \right)}}{{LY}_{\ell,m}\left( {\theta,\phi} \right)}\quad{where}}$$L = {\frac{1}{i}{\left( {r \times \nabla} \right).}}$
 87. The system ofclaim 85, wherein the coefficients α_(E)(l,m) and α_(M)(l,m) specify theamounts of electric (l,m) multipole and magnetic (l,m) multipole fields,and are determined by sources and boundary conditions as are therelative proportions in g_(l)(kr).
 88. The system of claim 86, whereinthe electric and magnetic coefficients from the sources is${a_{E}\left( {\ell,m} \right)} = {\frac{4\pi\quad k^{2}}{i\sqrt{\ell\left( {\ell + 1} \right)}}{\int{Y_{\ell}^{m^{*}}\left\{ {{\rho{\frac{\delta}{\delta\quad r}\left\lbrack {{rj}_{\ell}({kr})} \right\rbrack}} + {\frac{ik}{c}\left( {r \cdot J} \right){j_{\ell}({kr})}} - {{ik}{\nabla{\cdot \left( {r \times M} \right)}}{j_{\ell}({kr})}}} \right\}{\mathbb{d}^{3}x}}}}$and${a_{M}\left( {\ell,m} \right)} = {\frac{{- 4}\pi\quad k^{2}}{\sqrt{\ell\left( {\ell + 1} \right)}}{\int{{j_{\ell}({kr})}Y_{\ell}^{m^{*}}{L \cdot \left( {\frac{J}{c} + {\nabla{\times M}}} \right)}{\mathbb{d}^{3}x}}}}$respectively, where the distribution of charge ρ(x,t), current J(x,t),and intrinsic magnetization M(x,t) are harmonically varying sources:ρ(x)e^(−ω) ^(n) ^(t), J(x)e^(−ω) ^(n) ^(t), and M(x)e^(−ω) ^(n) ^(t).89. The system of claim 86, wherein the charge and intrinsicmagnetization terms are zero.
 90. The system of claim 87, wherein sincethe source dimensions are very small compared to a wavelength(kr_(max)<<1), the small argument limit can be used to give the magneticmultipole coefficient α_(M)(l,m) as${a_{M}\left( {\ell,m} \right)} = {{\frac{{- 4}\pi\quad k^{\ell + 2}}{\left( {{2\ell} + 1} \right)!!}\left( \frac{\ell + 1}{\ell} \right)^{1/2}\left( {M_{\ell\quad m} + M_{\ell\quad m}^{\prime}} \right)} = {\frac{{- 4}\pi\quad k^{\ell + 2}}{\frac{\left( {{2\ell} + 1} \right)!}{2^{n}{n!}}}\left( \frac{\ell + 1}{\ell} \right)^{1/2}\left( {M_{\ell\quad m} + M_{\ell\quad m}^{\prime}} \right)}}$where the magnetic multipole moments are$M_{\ell\quad m} = {{- \frac{1}{\ell + 1}}{\int{r^{\ell}Y_{\ell\quad m}^{*}{\nabla{\cdot \left( \frac{r \times J}{c} \right)}}{\mathbb{d}^{3}x}}}}$M_(ℓ  m)^(′) = −∫r^(ℓ)Y_(ℓ  m)^(*)∇⋅M𝕕³x.
 91. The system of claim 88,wherein the geometrical factor of the surface current-density functionof the orbitsphere about the z-axis is$\left( \frac{2}{3} \right)^{- 1}.$
 92. The system of claim 89, whereinthe multipole coefficient α_(Mag)(l,m) of the magnetic force is a Mag ⁡ (, m ) = 3 2 ( 2 ⁢ + 1 ) !! ⁢ 1 + 2 ⁢ ( + 1 ) 1 / 2 .
 93. The system ofclaim 90, wherein for singlet states with l≠0, a minimum energy isachieved with conservation of the photon's angular momentum of

when the magnetic moments of the corresponding angular momenta relativeto the electron velocity and corresponding Lorentzian forces superimposenegatively such that the spin component is radial (i_(r)-direction) andthe orbital component is central (−i_(r)-direction).
 94. The system ofclaim 91, wherein the amplitude of the orbital angular momentumL_(rotational orbital), is L = I ⁢   ⁢ ω ⁢   ⁢ i z = ℏ ⁡ [ ⁢ ( + 1 ) 2 + 2 ⁢ +1 ] 1 2 = ℏ ⁢ + 1 .
 95. The system of claim 92, wherein the magneticforce between the two electrons is F mag = - 1 n ⁢ 3 2 ( 2 ⁢ + 1 ) !! ⁢ ⁢1 + 2 ⁢ ( + 1 ) 1 / 2 ⁢ 1 2 ⁢ ℏ 2 m e ⁢ r 3 ⁢ ( s ⁡ ( s + 1 ) - + 1 ) . 96.The system of claim 93, wherein the force balance equation whichachieves the condition that the sum of the mechanical momentum andelectromagnetic momentum is m e ⁢ v 2 r 2 = ℏ 2 m e ⁢ r 2 3 = 1 n ⁢ ⅇ 2 4 ⁢πɛ o ⁢ r 2 2 - 1 n ⁢ 3 2 ( 2 ⁢ + 1 ) !! ⁢ ( + 1 ) 1 / 2 ⁢ 1 + 2 ⁢ 1 2 ⁢ ℏ 2 m e⁢r 3 ⁢ ( s ⁡ ( s + 1 ) - + 1 ) ; $s = {\frac{1}{2}.}$
 97. The system ofclaim 94, wherein r 2 = [ n + 3 4 ( 2 ⁢ + 1 ) !! ⁢ 1 + 2 ⁢ ( + 1 ) 1 / 2 ⁢ (3 4 - ⁢ + 1 ) ] ⁢ a He n = 2, 3, 4, …
 98. The system of claim 95, whereinthe excited-state energy is the energy stored in the electric field,E_(ele), which is the energy of the excited-state electron (electron 2)relative to the ionized electron at rest having zero energy:$E_{ele} = {{- \frac{1}{n}}{\frac{{\mathbb{e}}^{2}}{8{\pi ɛ}_{o}r_{2}}.}}$99. The system of claim 96, wherein using r₂, r₁ can be solved using theequal and opposite magnetic force of electron 2 on electron 1 and thecentral Coulombic force corresponding to the nuclear charge.
 100. Thesystem of claim 97, wherein the force balance between the centrifugaland electric and magnetic forces is m e ⁢ v 2 r 1 = ℏ 2 m e ⁢ r 1 3 = 2 ⁢ ⅇ2 4 ⁢ πɛ o ⁢ r 1 2 + 1 n ⁢ 3 2 ( 2 ⁢ + 1 ) !! ⁢ ( + 1 ) 1 / 2 ⁢ 1 + 2 ⁢ 1 2 ⁢ ℏ2 m e ⁢ r 2 3 ⁢ ( s ⁡ ( s + 1 ) - + 1 ) such that with ${s = \frac{1}{2}},$${r_{1}^{3} + {\frac{n\quad 8r_{1}r_{2}^{3}}{3\left( {\sqrt{\frac{3}{4}} - \sqrt{\frac{\ell}{\ell + 1}}} \right)}{\left( {{2\ell} + 1} \right)!!}\left( \frac{\ell}{\ell + 1} \right)^{1/2}\left( {\ell + 2} \right)} - {\frac{n\quad 4r_{2}^{3}}{3\left( {\sqrt{\frac{3}{4}} - \sqrt{\frac{\ell}{\ell + 1}}} \right)}{\left( {{2\ell} + 1} \right)!!}\left( \frac{\ell}{\ell + 1} \right)^{1/2}\left( {\ell + 2} \right)}} = 0$n = 2, 3, 4, …$r_{1} = {r_{13} = {\frac{1}{2} + \frac{1}{16g} + {O\left( g^{{- 3}/2} \right)}}}$where r₁ and r₂ are in units of α_(He).
 101. The system of claim 98,wherein for the triplet-excited state with l≠0, a minimum energy isachieved with conservation of the photon's angular momentum of

when the magnetic moments of the corresponding angular momentasuperimpose negatively such that the spin component is central and theorbital component is radial.
 102. The system of claim 99, wherein thespin is doubled such that the force balance equation is given by m e ⁢ v2 r 2 = ℏ 2 m e ⁢ r 2 3 = 1 n ⁢ ⅇ 2 4 ⁢ πɛ o ⁢ r 2 2 + 1 n ⁢ 3 2 ( 2 ⁢ + 1 )!! ⁢ ( + 1 ) 1 / 2 ⁢ 1 + 2 ⁢ 1 2 ⁢ ℏ 2 m e ⁢ r 3 ⁢ ( 2 ⁢ s ⁡ ( s + 1 ) - + 1 ) ;$\quad{s = {\frac{1}{2}.}}\quad$
 103. The system of claim 100, wherein$r_{2} = {\left\lbrack {n - {\frac{\frac{3}{4}}{\left( {{2\ell} + 1} \right)!!}\frac{1}{\ell + 2}\left( \frac{\ell + 1}{\ell} \right)^{1/2}\left( {{2\sqrt{\frac{3}{4}}} - \sqrt{\frac{\ell}{\ell + 1}}} \right)}} \right\rbrack a_{He}}$n = 2, 3, 4, …
 104. The system of claim 101, wherein the excited-stateenergy is the energy stored in the electric field, E_(ele), which is theenergy of the excited-state electron (electron 2) relative to theionized electron at rest having zero energy:$E_{ele} = {{- \frac{1}{n}}{\frac{e^{2}}{8\quad{\pi ɛ}_{0}r_{2}}.}}$105. The system of claim 102, wherein using r₂, r₁ can be solved usingthe equal and opposite magnetic force of electron 2 on electron 1 andthe central Coulombic force corresponding to the nuclear charge. 106.The system of claim 103, wherein the force balance between thecentrifugal and electric and magnetic forces is$\frac{m_{e}v^{2}}{r_{1}} = {\frac{\hslash^{2}}{m_{e}r_{1}^{3}} = {\frac{2e^{2}}{4{\pi ɛ}_{o}r_{1}^{2}} - {\frac{1}{n}\frac{\frac{3}{2}}{\left( {{2\ell} + 1} \right)!!}\left( \frac{\ell + 1}{\ell} \right)^{1/2}\frac{1}{\ell + 2}\frac{1}{2}\frac{\hslash}{m_{e}r_{2}^{3}}\left( {{2\sqrt{s\left( {s + 1} \right)}} - \sqrt{\frac{\ell}{\ell + 1}}} \right)}}}$such that with ${s = \frac{1}{2}},$${r_{1}^{3} - {\frac{n\quad 8r_{1}r_{2}^{3}}{3\left( {\sqrt{\frac{3}{4}} - \sqrt{\frac{\ell}{\ell + 1}}} \right)}{\left( {{2\ell} + 1} \right)!!}\left( \frac{\ell}{\ell + 1} \right)^{1/2}\left( {\ell + 2} \right)} + {\frac{n\quad 4r_{2}^{3}}{3\left( {\sqrt{\frac{3}{4}} - \sqrt{\frac{\ell}{\ell + 1}}} \right)}{\left( {{2\ell} + 1} \right)!!}\left( \frac{\ell}{\ell + 1} \right)^{1/2}\left( {\ell + 2} \right)}} = 0$${n = 2},3,4,{{\ldots r_{1}} = {r_{13} = {{- \sqrt{\frac{2}{3}g}}\left( {{\cos\frac{\theta}{3}} - {\sqrt{3}\sin\frac{\theta}{3}}} \right)}}},$where r₁ and r₂ are in units of α_(He).
 107. The system of claim 104,wherein the spin-orbital coupling force is used in the force balanceequation wherein the corresponding energy E_(s/o) is given by$E_{s/o} = {{2\frac{\alpha}{2\pi}\left( \frac{e\quad\hslash}{2m_{e}} \right)\frac{\mu_{0}e\quad\hslash}{2\left( {2\pi\quad m_{e}} \right)\left( \frac{r}{2\pi} \right)^{3}}\sqrt{\frac{3}{4}}} = {\frac{\alpha\quad\pi\quad\mu_{0}e^{2}\hslash^{2}}{m_{e}^{2}r^{3}}{\sqrt{\frac{3}{4}}.}}}$108. The system of claim 105, wherein the force balance also includes aterm corresponding to the frequency shift derived after that of the Lambshift.
 109. The system of claim 106, wherein with Δm_(l)=−1 is includedand the energy, E_(FS), for the ²P_(3/2)→²P_(1/2) transition called thefine structure splitting is given by: $\begin{matrix}{E_{FS} = {{\frac{{\alpha^{5}\left( {2\pi} \right)}^{2}}{8}m_{e}c^{2}\sqrt{\frac{3}{4}}} + \left( {13.5983\quad{{eV}\left( {1 - \frac{1}{2^{2}}} \right)}} \right)^{2}}} \\{\left\lbrack {\frac{\left( {\frac{3}{4\pi}\left( {1 - \sqrt{\frac{3}{4}}} \right)} \right)^{2}}{2h\quad\mu_{e}c^{2}} + \frac{\left( {1 + \left( {1 - \sqrt{\frac{3}{4}}} \right)} \right)^{2}}{2{hm}_{H}c^{2}}} \right\rbrack} \\{= {{4.5190 \times 10^{- 5}\quad{eV}} + {1.75407 \times 10^{- 7}\quad{eV}}}} \\{= {4.53659 \times 10^{- 5}\quad{eV}}}\end{matrix}$ where the first term corresponds to E_(s/o) expressed interms of the mass energy of the electron and the second and third termscorrespond to the electron recoil and atom recoil, respectively.
 110. Amethod comprising the steps of; a.) inputting the electron functionsthat obey Maxwell's equations; b.) determining the correspondingcentrifugal, Coulombic, diamagnetic and paramagnetic forces for a givenset of quantum numbers corresponding to a solution of Maxwell'sequations for at least one photon and one electron of the excited state;d.) forming the force balance equation comprising the centrifugal forceequal to the sum of the Coulombic, diamagnetic and paramagnetic forces;e.) solving the force balance equation for the electron radii; f.)calculating the energy of the electrons using the radii and thecorresponding electric and magnetic energies, and g.) outputting thecalculated energy of the electrons.
 111. The method according to claim110, further comprising displaying the electron function at each radiusto visualize the excited-state atom or atomic ion.
 112. The methodaccording to claim 110, wherein the output is rendered using theelectron functions.
 113. The method according to claim 110, wherein theelectron functions are given by at least one of the group comprising:ℓ = 0${\rho\left( {r,\theta,\phi,t} \right)} = {{\frac{e}{8\pi\quad r^{2}}\left\lbrack {\delta\left( {r - r_{n}} \right)} \right\rbrack}\left\lbrack {{Y_{0}^{0}\left( {\theta,\phi} \right)} + {Y_{\ell}^{m}\left( {\theta,\phi} \right)}} \right\rbrack}$ℓ ≠ 0${\rho\left( {r,\theta,\phi,t} \right)} = {{\frac{e}{4\pi\quad r^{2}}\left\lbrack {\delta\left( {r - r_{n}} \right)} \right\rbrack}\left\lbrack {{Y_{0}^{0}\left( {\theta,\phi} \right)} + {{Re}\left\{ {{Y_{\ell}^{m}\left( {\theta,\phi} \right)}{\mathbb{e}}^{{\mathbb{i}\omega}_{n}t}} \right\}}} \right\rbrack}$where Y_(l) ^(m)(θ,φ) are the spherical harmonic functions that spinabout the z-axis with angular frequency ω_(n) with Y₀ ⁰(θ,φ) theconstant function. Re{Y_(l) ^(m)(θ,φ)e^(iωj)}=P_(l) ^(m)(cosθ)cos(mφ+{dot over (ω)}_(n)t) where to keep the form of the sphericalharmonic as a traveling wave about the z-axis, ω_(n)=mω_(n).
 114. Themethod according to claim 113, wherein the force balance equation isgiven by at least one of the group comprising:$\frac{m_{e}v^{2}}{r_{2}} = {\frac{\hslash^{2}}{m_{e}r_{2}^{3}} = {{\frac{1}{n}\frac{e^{2}}{4\pi\quad ɛ_{o}r_{2}^{2}}} + {\frac{2}{3}\frac{1}{n}\frac{\hslash^{2}}{2m_{e}r_{2}^{3}}\sqrt{s\left( {s + 1} \right)}}}}$$\frac{m_{e}v^{2}}{r_{1}} = {\frac{\hslash^{2}}{m_{e}r_{1}^{3}} = {\frac{2e^{2}}{4\pi\quad ɛ_{o}r_{1}^{2}} - {\frac{1}{3n}\frac{\hslash^{2}}{m_{e}r_{2}^{3}}\sqrt{s\left( {s + 1} \right)}}}}$$\frac{m_{e}v^{2}}{r_{2}} = {\frac{\hslash^{2}}{m_{e}r_{2}^{3}} = {{\frac{1}{n}\frac{e^{2}}{4\pi\quad ɛ_{o}r_{2}^{2}}} + {2\frac{2}{3}\frac{1}{n}\frac{\hslash^{2}}{2m_{e}r_{2}^{3}}\sqrt{s\left( {s + 1} \right)}}}}$$\frac{m_{e}v^{2}}{r_{1}} = {\frac{\hslash^{2}}{m_{e}r_{1}^{3}} = {\frac{2e^{2}}{4\pi\quad ɛ_{o}r_{1}^{2}} - {\frac{2}{3n}\frac{\hslash^{2}}{m_{e}r_{2}^{3}}\sqrt{s\left( {s + 1} \right)}}}}$$\begin{matrix}{\frac{m_{e}v^{2}}{r_{2}} = {\frac{\hslash^{2}}{m_{e}r_{2}^{3}} = {{\frac{1}{n}\frac{e^{2}}{4\pi\quad ɛ_{o}r_{2}^{2}}} - {\frac{1}{n}\frac{\frac{3}{2}}{\left( {{2\ell} + 1} \right)!!}\left( \frac{\ell + 1}{\ell} \right)^{1/2}}}}} \\{\frac{1}{{\ell + 2}\quad}\frac{1}{2}\frac{\hslash^{2}}{m_{e}r^{3}}\left( {\sqrt{s\left( {s + 1} \right)} - \sqrt{\frac{\ell}{\ell + 1}}} \right)}\end{matrix}$ $\begin{matrix}{\frac{m_{e}v^{2}}{r_{1}} = {\frac{\hslash^{2}}{m_{e}r_{1}^{3}} = {\frac{2e^{2}}{4\pi\quad ɛ_{o}r_{1}^{2}} + {\frac{1}{n}\frac{\frac{3}{2}}{\left( {{2\ell} + 1} \right)!!}\left( \frac{\ell + 1}{\ell} \right)^{1/2}}}}} \\{\frac{1}{{\ell + 2}\quad}\frac{1}{2}\frac{\hslash^{2}}{m_{e}r_{2}^{3}}\left( {\sqrt{s\left( {s + 1} \right)} - \sqrt{\frac{\ell}{\ell + 1}}} \right)}\end{matrix}$ $\begin{matrix}{\frac{m_{e}v^{2}}{r_{2}} = {\frac{\hslash^{2}}{m_{e}r_{2}^{3}} = {{\frac{1}{n}\frac{e^{2}}{4\pi\quad ɛ_{o}r_{2}^{2}}} + {\frac{1}{n}\frac{\frac{3}{2}}{\left( {{2\ell} + 1} \right)!!}\left( \frac{\ell + 1}{\ell} \right)^{1/2}}}}} \\{\frac{1}{{\ell + 2}\quad}\frac{1}{2}\frac{\hslash^{2}}{m_{e}r^{3}}\left( {{2\sqrt{s\left( {s + 1} \right)}} - \sqrt{\frac{\ell}{\ell + 1}}} \right)}\end{matrix}$ $\begin{matrix}{\frac{m_{e}v^{2}}{r_{1}} = {\frac{\hslash^{2}}{m_{e}r_{1}^{3}} = {\frac{2e^{2}}{4\pi\quad ɛ_{o}r_{1}^{2}} - {\frac{1}{n}\frac{\frac{3}{2}}{\left( {{2\ell} + 1} \right)!!}\left( \frac{\ell + 1}{\ell} \right)^{1/2}}}}} \\{\frac{1}{{\ell + 2}\quad}\frac{1}{2}\frac{\hslash^{2}}{m_{e}r_{2}^{3}}\left( {{2\sqrt{s\left( {s + 1} \right)}} - \sqrt{\frac{\ell}{\ell + 1}}} \right)}\end{matrix}$ wherein $s = {\frac{1}{2}.}$
 115. The method according toclaim 113, wherein the force is given by at least one of the forces ofthe force balance equations of the group comprising:$\frac{m_{e}v^{2}}{r_{2}} = {\frac{\hslash^{2}}{m_{e}r_{2}^{3}} = {{\frac{1}{n}\frac{e^{2}}{4\pi\quad ɛ_{o}r_{2}^{2}}} + {\frac{2}{3}\frac{1}{n}\frac{\hslash^{2}}{2m_{e}r_{2}^{3}}\sqrt{s\left( {s + 1} \right)}}}}$$\frac{m_{e}v^{2}}{r_{1}} = {\frac{\hslash^{2}}{m_{e}r_{1}^{3}} = {\frac{2e^{2}}{4\pi\quad ɛ_{o}r_{1}^{2}} - {\frac{1}{3n}\frac{\hslash^{2}}{m_{e}r_{2}^{3}}\sqrt{s\left( {s + 1} \right)}}}}$$\frac{m_{e}v^{2}}{r_{2}} = {\frac{\hslash^{2}}{m_{e}r_{2}^{3}} = {{\frac{1}{n}\frac{e^{2}}{4\pi\quad ɛ_{o}r_{2}^{2}}} + {2\frac{2}{3}\frac{1}{n}\frac{\hslash^{2}}{2m_{e}r_{2}^{3}}\sqrt{s\left( {s + 1} \right)}}}}$$\frac{m_{e}v^{2}}{r_{1}} = {\frac{\hslash^{2}}{m_{e}r_{1}^{3}} = {\frac{2e^{2}}{4\pi\quad ɛ_{o}r_{1}^{2}} - {\frac{2}{3n}\frac{\hslash^{2}}{m_{e}r_{2}^{3}}\sqrt{s\left( {s + 1} \right)}}}}$$\begin{matrix}{\frac{m_{e}v^{2}}{r_{2}} = {\frac{\hslash^{2}}{m_{e}r_{2}^{3}} = {{\frac{1}{n}\frac{e^{2}}{4\pi\quad ɛ_{o}r_{2}^{2}}} - {\frac{1}{n}\frac{\frac{3}{2}}{\left( {{2\ell} + 1} \right)!!}\left( \frac{\ell + 1}{\ell} \right)^{1/2}}}}} \\{\frac{1}{{\ell + 2}\quad}\frac{1}{2}\frac{\hslash^{2}}{m_{e}r^{3}}\left( {\sqrt{s\left( {s + 1} \right)} - \sqrt{\frac{\ell}{\ell + 1}}} \right)}\end{matrix}$ $\begin{matrix}{\frac{m_{e}v^{2}}{r_{1}} = {\frac{\hslash^{2}}{m_{e}r_{1}^{3}} = {\frac{2e^{2}}{4\pi\quad ɛ_{o}r_{1}^{2}} + {\frac{1}{n}\frac{\frac{3}{2}}{\left( {{2\ell} + 1} \right)!!}\left( \frac{\ell + 1}{\ell} \right)^{1/2}}}}} \\{\frac{1}{{\ell + 2}\quad}\frac{1}{2}\frac{\hslash^{2}}{m_{e}r_{2}^{3}}\left( {\sqrt{s\left( {s + 1} \right)} - \sqrt{\frac{\ell}{\ell + 1}}} \right)}\end{matrix}$ $\begin{matrix}{\frac{m_{e}v^{2}}{r_{2}} = {\frac{\hslash^{2}}{m_{e}r_{2}^{3}} = {{\frac{1}{n}\frac{e^{2}}{4\pi\quad ɛ_{o}r_{2}^{2}}} + {\frac{1}{n}\frac{\frac{3}{2}}{\left( {{2\ell} + 1} \right)!!}\left( \frac{\ell + 1}{\ell} \right)^{1/2}}}}} \\{\frac{1}{{\ell + 2}\quad}\frac{1}{2}\frac{\hslash^{2}}{m_{e}r^{3}}\left( {{2\sqrt{s\left( {s + 1} \right)}} - \sqrt{\frac{\ell}{\ell + 1}}} \right)}\end{matrix}$ $\begin{matrix}{\frac{m_{e}v^{2}}{r_{1}} = {\frac{\hslash^{2}}{m_{e}r_{1}^{3}} = {\frac{2e^{2}}{4\pi\quad ɛ_{o}r_{1}^{2}} - {\frac{1}{n}\frac{\frac{3}{2}}{\left( {{2\ell} + 1} \right)!!}\left( \frac{\ell + 1}{\ell} \right)^{1/2}}}}} \\{\frac{1}{{\ell + 2}\quad}\frac{1}{2}\frac{\hslash^{2}}{m_{e}r_{2}^{3}}\left( {{2\sqrt{s\left( {s + 1} \right)}} - \sqrt{\frac{\ell}{\ell + 1}}} \right)}\end{matrix}\quad$ ${{wherein}\quad s} = {\frac{1}{2}.}$
 116. The methodaccording to claim 110, wherein the radii are given by at least one ofthe group comprising:${r_{2} = {{\left\lbrack {n - \frac{\sqrt{\frac{3}{4}}}{3}} \right\rbrack a_{He}\quad n} = 2}},3,4,\ldots$$r_{1} = {r_{13} = {{- \sqrt{\frac{2}{3}}}{g\left( {{\cos\frac{\theta}{3}} - {\sqrt{3}\sin\frac{\theta}{3}}} \right)}}}$${r_{2} = {{\left\lbrack {n - \frac{2\sqrt{\frac{3}{4}}}{3}} \right\rbrack a_{He}\quad n} = 2}},3,4,\ldots$$r_{1} = {r_{13} = {{- \sqrt{\frac{2}{3}}}{g\left( {{\cos\frac{\theta}{3}} - {\sqrt{3}\sin\frac{\theta}{3}}} \right)}}}$$\begin{matrix}{r_{2} = {\left\lbrack {n + {\frac{\frac{3}{4}}{\left( {{2\ell} + 1} \right)!!}\frac{1}{\ell + 2}\left( \frac{\ell + 1}{\ell} \right)^{1/2}\left( {\sqrt{\frac{3}{4}} - \sqrt{\frac{\ell}{\ell + 1}}} \right)}} \right\rbrack a_{He}}} \\{{n = 2},3,4,\ldots}\end{matrix}$$r_{1} = {r_{11} = {\sqrt[3]{- \frac{g}{2}}\left\{ {\sqrt[3]{1 + \sqrt{1 - {\frac{32}{27}g}}} - \sqrt[3]{\sqrt{1 - {\frac{32}{27}g}} - 1}} \right\}}}$$\begin{matrix}{r_{2} = {\left\lbrack {n - {\frac{\frac{3}{4}}{\left( {{2\ell} + 1} \right)!!}\frac{1}{\ell + 2}\left( \frac{\ell + 1}{\ell} \right)^{1/2}\left( {{2\sqrt{\frac{3}{4}}} - \sqrt{\frac{\ell}{\ell + 1}}} \right)}} \right\rbrack a_{He}}} \\{{n = 2},3,4,\ldots}\end{matrix}$$r_{1} = {r_{13} = {{- \sqrt{\frac{2}{3}g}}\left( {{\cos\frac{\theta}{3}} - {\sqrt{3}\sin\frac{\theta}{3}}} \right)}}$where r₁ and r₂ are in units of α_(He).
 117. The method according toclaim 110, wherein the electric energy of each electron of radius r_(n)is given by:$E_{ele} = {{- \frac{1}{n}}\frac{e^{2}}{8\pi\quad ɛ_{o}r_{2}}}$